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11.2: Correlation Hypothesis Test

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The correlation coefficient, $r$, tells us about the strength and direction of the linear relationship between $x$ and $y$. However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient $r$ and the sample size $n$, together. We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute $r$, the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, $r$, is our estimate of the unknown population correlation coefficient.

The symbol for the population correlation coefficient is $\rho$, the Greek letter "rho."
$\rho =$ population correlation coefficient (unknown)
$r =$ sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient $\rho$ is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient $r$ and the sample size $n$.

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant."

Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from zero.
What the conclusion means: There is a significant linear relationship between $x$ and $y$. We can use the regression line to model the linear relationship between $x$ and $y$ in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant".

Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is not significantly different from zero."
What the conclusion means: There is not a significant linear relationship between $x$ and $y$. Therefore, we CANNOT use the regression line to model a linear relationship between $x$ and $y$ in the population.
If $r$ is significant and the scatter plot shows a linear trend, the line can be used to predict the value of $y$ for values of $x$ that are within the domain of observed $x$ values.
If $r$ is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
If $r$ is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed $x$ values in the data.

PERFORMING THE HYPOTHESIS TEST

Null Hypothesis: $H_{0}: \rho = 0$
Alternate Hypothesis: $H_{a}: \rho \neq 0$

WHAT THE HYPOTHESES MEAN IN WORDS:

Null Hypothesis $H_{0}$ : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between $x$ and $y$ in the population.
Alternate Hypothesis $H_{a}$ : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between $x$ and $y$ in the population.

DRAWING A CONCLUSION:There are two methods of making the decision. The two methods are equivalent and give the same result.

Method 1: Using the $p\text{-value}$
Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, $\alpha = 0.05$

Using the $p\text{-value}$ method, you could choose any appropriate significance level you want; you are not limited to using $\alpha = 0.05$. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, $\alpha = 0.05$. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

METHOD 1: Using a $p\text{-value}$ to make a decision

Using the ti83, 83+, 84, 84+ calculator.

To calculate the $p\text{-value}$ using LinRegTTEST:

On the LinRegTTEST input screen, on the line prompt for $\beta$ or $\rho$, highlight "$\neq 0$"

The output screen shows the $p\text{-value}$ on the line that reads "$p =$".

(Most computer statistical software can calculate the $p\text{-value}$.)

If the $p\text{-value}$ is less than the significance level ( $\alpha = 0.05$ ):

Decision: Reject the null hypothesis.
Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is significantly different from zero."

If the $p\text{-value}$ is NOT less than the significance level ( $\alpha = 0.05$ )

Decision: DO NOT REJECT the null hypothesis.
Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between $x$ and $y$ because the correlation coefficient is NOT significantly different from zero."

Calculation Notes:

You will use technology to calculate the $p\text{-value}$. The following describes the calculations to compute the test statistics and the $p\text{-value}$:
The $p\text{-value}$ is calculated using a $t$-distribution with $n - 2$ degrees of freedom.
The formula for the test statistic is $t = \frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}$. The value of the test statistic, $t$, is shown in the computer or calculator output along with the $p\text{-value}$. The test statistic $t$ has the same sign as the correlation coefficient $r$.
The $p\text{-value}$ is the combined area in both tails.

An alternative way to calculate the $p\text{-value}$ ( $p$ ) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: $p\text{-value}$ method

Consider the third exam/final exam example.
The line of best fit is: $\hat{y} = -173.51 + 4.83x$ with $r = 0.6631$ and there are $n = 11$ data points.
Can the regression line be used for prediction? Given a third exam score ( $x$ value), can we use the line to predict the final exam score (predicted $y$ value)?
$H_{0}: \rho = 0$
$H_{a}: \rho \neq 0$
$\alpha = 0.05$
The $p\text{-value}$ is 0.026 (from LinRegTTest on your calculator or from computer software).
The $p\text{-value}$, 0.026, is less than the significance level of $\alpha = 0.05$.
Decision: Reject the Null Hypothesis $H_{0}$
Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ($x$) and the final exam score ($y$) because the correlation coefficient is significantly different from zero.

Because $r$ is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of $r$ is significant or not . Compare $r$ to the appropriate critical value in the table. If $r$ is not between the positive and negative critical values, then the correlation coefficient is significant. If $r$ is significant, then you may want to use the line for prediction.

Example $\PageIndex{1}$

Suppose you computed $r = 0.801$ using $n = 10$ data points. $df = n - 2 = 10 - 2 = 8$. The critical values associated with $df = 8$ are $-0.632$ and $+0.632$. If $r <$ negative critical value or $r >$ positive critical value, then $r$ is significant. Since $r = 0.801$ and $0.801 > 0.632$, $r$ is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

Horizontal number line with values of -1, -0.632, 0, 0.632, 0.801, and 1. A dashed line above values -0.632, 0, and 0.632 indicates not significant values.

Exercise $\PageIndex{1}$

For a given line of best fit, you computed that $r = 0.6501$ using $n = 12$ data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because $r >$ the positive critical value.

Example $\PageIndex{2}$

Suppose you computed $r = –0.624$ with 14 data points. $df = 14 – 2 = 12$. The critical values are $-0.532$ and $0.532$. Since $-0.624 < -0.532$, $r$ is significant and the line can be used for prediction

Horizontal number line with values of -0.624, -0.532, and 0.532.

Exercise $\PageIndex{2}$

For a given line of best fit, you compute that $r = 0.5204$ using $n = 9$ data points, and the critical value is $0.666$. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because $r <$ the positive critical value.

Example $\PageIndex{3}$

Suppose you computed $r = 0.776$ and $n = 6$. $df = 6 - 2 = 4$. The critical values are $-0.811$ and $0.811$. Since $-0.811 < 0.776 < 0.811$, $r$ is not significant, and the line should not be used for prediction.

Horizontal number line with values -0.924, -0.532, and 0.532.

Exercise $\PageIndex{3}$

For a given line of best fit, you compute that $r = -0.7204$ using $n = 8$ data points, and the critical value is $= 0.707$. Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because $r <$ the negative critical value.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example. The line of best fit is: $\hat{y} = -173.51 + 4.83x$ with $r = 0.6631$ and there are $n = 11$ data points. Can the regression line be used for prediction? Given a third-exam score ( $x$ value), can we use the line to predict the final exam score (predicted $y$ value)?

Use the "95% Critical Value" table for $r$ with $df = n - 2 = 11 - 2 = 9$.
The critical values are $-0.602$ and $+0.602$
Since $0.6631 > 0.602$, $r$ is significant.
Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ($x$) and the final exam score ($y$) because the correlation coefficient is significantly different from zero.

Example $\PageIndex{4}$

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if $r$ is significant and the line of best fit associated with each r can be used to predict a $y$ value. If it helps, draw a number line.

$r = –0.567$ and the sample size, $n$, is $19$. The $df = n - 2 = 17$. The critical value is $-0.456$. $-0.567 < -0.456$ so $r$ is significant.
$r = 0.708$ and the sample size, $n$, is $9$. The $df = n - 2 = 7$. The critical value is $0.666$. $0.708 > 0.666$ so $r$ is significant.
$r = 0.134$ and the sample size, $n$, is $14$. The $df = 14 - 2 = 12$. The critical value is $0.532$. $0.134$ is between $-0.532$ and $0.532$ so $r$ is not significant.
$r = 0$ and the sample size, $n$, is five. No matter what the $dfs$ are, $r = 0$ is between the two critical values so $r$ is not significant.

Exercise $\PageIndex{4}$

For a given line of best fit, you compute that $r = 0$ using $n = 100$ data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between $x$ and $y$ in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between $x$ and $y$ in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

The assumptions underlying the test of significance are:

There is a linear relationship in the population that models the average value of $y$ for varying values of $x$. In other words, the expected value of $y$ for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
The $y$ values for any particular $x$ value are normally distributed about the line. This implies that there are more $y$ values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of $y$ values lie on the line.
The standard deviations of the population $y$ values about the line are equal for each value of $x$. In other words, each of these normal distributions of $y$ values has the same shape and spread about the line.
The residual errors are mutually independent (no pattern).
The data are produced from a well-designed, random sample or randomized experiment.

The left graph shows three sets of points. Each set falls in a vertical line. The points in each set are normally distributed along the line — they are densely packed in the middle and more spread out at the top and bottom. A downward sloping regression line passes through the mean of each set. The right graph shows the same regression line plotted. A vertical normal curve is shown for each line.

Linear regression is a procedure for fitting a straight line of the form $\hat{y} = a + bx$ to data. The conditions for regression are:

Linear In the population, there is a linear relationship that models the average value of $y$ for different values of $x$.
Independent The residuals are assumed to be independent.
Normal The $y$ values are distributed normally for any value of $x$.
Equal variance The standard deviation of the $y$ values is equal for each $x$ value.
Random The data are produced from a well-designed random sample or randomized experiment.

The slope $b$ and intercept $a$ of the least-squares line estimate the slope $\beta$ and intercept $\alpha$ of the population (true) regression line. To estimate the population standard deviation of $y$, $\sigma$, use the standard deviation of the residuals, $s$. $s = \sqrt{\frac{SEE}{n-2}}$. The variable $\rho$ (rho) is the population correlation coefficient. To test the null hypothesis $H_{0}: \rho =$ hypothesized value , use a linear regression t-test. The most common null hypothesis is $H_{0}: \rho = 0$ which indicates there is no linear relationship between $x$ and $y$ in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

Formula Review

Least Squares Line or Line of Best Fit:

\[\hat{y} = a + bx\]

\[a = y\text{-intercept}\]

\[b = \text{slope}\]

Standard deviation of the residuals:

\[s = \sqrt{\frac{SSE}{n-2}}\]

\[SSE = \text{sum of squared errors}\]

\[n = \text{the number of data points}\]

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Chapter 12.5: Testing the Significance of the Correlation Coefficient

The correlation coefficient, r , tells us about the strength and direction of the linear relationship between x and y . However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient r and the sample size n , together.

We perform a hypothesis test of the “significance of the correlation coefficient” to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute r , the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only have sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, r , is our estimate of the unknown population correlation coefficient.

The symbol for the population correlation coefficient is ρ , the Greek letter “rho.”
ρ = population correlation coefficient (unknown)
r = sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient ρ is “close to zero” or “significantly different from zero”. We decide this based on the sample correlation coefficient r and the sample size n .

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.”

Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.
What the conclusion means: There is a significant linear relationship between x and y . We can use the regression line to model the linear relationship between x and y in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant”.

Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.”
What the conclusion means: There is not a significant linear relationship between x and y . Therefore, we CANNOT use the regression line to model a linear relationship between x and y in the population.
If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

PERFORMING THE HYPOTHESIS TEST

Null Hypothesis: H 0 : ρ = 0
Alternate Hypothesis: H a : ρ ≠ 0

WHAT THE HYPOTHESES MEAN IN WORDS:

Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

DRAWING A CONCLUSION: There are two methods of making the decision. The two methods are equivalent and give the same result.

Method 1: Using the p -value
Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05

Using the p -value method, you could choose any appropriate significance level you want; you are not limited to using α = 0.05. But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, α = 0.05. (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

METHOD 1: Using a p -value to make a decision

To calculate the p -value using LinRegTTEST: On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight “ ≠ 0 “ The output screen shows the p-value on the line that reads “p =”. (Most computer statistical software can calculate the p -value.)

Decision: Reject the null hypothesis.
Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”
Decision: DO NOT REJECT the null hypothesis.
Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.”
You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
The p -value is calculated using a t -distribution with n – 2 degrees of freedom.

$t=\frac{r\sqrt{n-2}}{\sqrt{1-{r}^{2}}}$

The p -value is the combined area in both tails.

An alternative way to calculate the p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

Consider the third exam/final exam example .
The line of best fit is: ŷ = -173.51 + 4.83 x with r = 0.6631 and there are n = 11 data points.
Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
H 0 : ρ = 0

H a : ρ ≠ 0

The p -value is 0.026 (from LinRegTTest on your calculator or from computer software).
The p -value, 0.026, is less than the significance level of α = 0.05.
Decision: Reject the Null Hypothesis H 0
Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

Suppose you computed r = 0.801 using n = 10 data points. df = n – 2 = 10 – 2 = 8. The critical values associated with df = 8 are -0.632 and + 0.632. If r < negative critical value or r > positive critical value, then r issignificant. Since r = 0.801 and 0.801 > 0.632, r is significant and the line may be usedfor prediction. If you view this example on a number line, it will help you.

For a given line of best fit, you computed that r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

Suppose you computed r = –0.624 with 14 data points. df = 14 – 2 = 12. The critical values are –0.532 and 0.532. Since –0.624 < –0.532, r is significant and the line can be used for prediction

For a given line of best fit, you compute that r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

Suppose you computed r = 0.776 and n = 6. df = 6 – 2 = 4. The critical values are –0.811 and 0.811. Since –0.811 < 0.776 < 0.811, r is not significant, and the line should not be used for prediction.

For a given line of best fit, you compute that r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example . The line of best fit is: ŷ = –173.51+4.83 x with r = 0.6631 and there are n = 11 data points. Can the regression line be used for prediction? Given a third-exam score ( x value), can we use the line to predict the final exam score (predicted y value)?

Use the “95% Critical Value” table for r with df = n – 2 = 11 – 2 = 9.
The critical values are –0.602 and +0.602
Since 0.6631 > 0.602, r is significant.
Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if r is significant and the line of best fit associated with each r can be used to predict a y value. If it helps, draw a number line.

r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
The residual errors are mutually independent (no pattern).
The data are produced from a well-designed, random sample or randomized experiment.

Chapter Review

Linear regression is a procedure for fitting a straight line of the form ŷ = a + bx to data. The conditions for regression are:

Linear In the population, there is a linear relationship that models the average value of y for different values of x .
Independent The residuals are assumed to be independent.
Normal The y values are distributed normally for any value of x .
Equal variance The standard deviation of the y values is equal for each x value.
Random The data are produced from a well-designed random sample or randomized experiment.

$s=\sqrt{\frac{SEE}{n-2}}$

Formula Review

Least Squares Line or Line of Best Fit:

$\stackrel{^}{y}=a+bx$

a = y -intercept

Standard deviation of the residuals:

$s=\sqrt{\frac{SEE}{n-2}}.$

SSE = sum of squared errors

n = the number of data points

When testing the significance of the correlation coefficient, what is the null hypothesis?

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

If the level of significance is 0.05 and the p -value is 0.04, what conclusion can you draw?

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Linear Regression and Correlation

Testing the Significance of the Correlation Coefficient

OpenStaxCollege

[latexpage]

The symbol for the population correlation coefficient is ρ , the Greek letter “rho.”
ρ = population correlation coefficient (unknown)
r = sample correlation coefficient (known; calculated from sample data)

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is “significant.”

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is “not significant”.

If r is significant and the scatter plot shows a linear trend, the line can be used to predict the value of y for values of x that are within the domain of observed x values.
If r is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
If r is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed x values in the data.

PERFORMING THE HYPOTHESIS TEST

Null Hypothesis: H 0 : ρ = 0
Alternate Hypothesis: H a : ρ ≠ 0

WHAT THE HYPOTHESES MEAN IN WORDS:

Null Hypothesis H 0 : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between x and y in the population.
Alternate Hypothesis H a : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between x and y in the population.

DRAWING A CONCLUSION: There are two methods of making the decision. The two methods are equivalent and give the same result.

Method 1: Using the p -value
Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, α = 0.05

METHOD 1: Using a p -value to make a decision

To calculate the p -value using LinRegTTEST:

On the LinRegTTEST input screen, on the line prompt for β or ρ , highlight “ ≠ 0 “

The output screen shows the p-value on the line that reads “p =”.

Decision: Reject the null hypothesis.
Conclusion: “There is sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is significantly different from zero.”
Decision: DO NOT REJECT the null hypothesis.
Conclusion: “There is insufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is NOT significantly different from zero.”
You will use technology to calculate the p -value. The following describes the calculations to compute the test statistics and the p -value:
The p -value is calculated using a t -distribution with n – 2 degrees of freedom.
The formula for the test statistic is $t=\frac{r\sqrt{n-2}}{\sqrt{1-{r}^{2}}}$. The value of the test statistic, t , is shown in the computer or calculator output along with the p -value. The test statistic t has the same sign as the correlation coefficient r .
The p -value is the combined area in both tails.

An alternative way to calculate the p -value (p) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

Consider the third exam/final exam example .
The line of best fit is: ŷ = -173.51 + 4.83 x with r = 0.6631 and there are n = 11 data points.
Can the regression line be used for prediction? Given a third exam score ( x value), can we use the line to predict the final exam score (predicted y value)?
H 0 : ρ = 0

H a : ρ ≠ 0

The p -value is 0.026 (from LinRegTTest on your calculator or from computer software).
The p -value, 0.026, is less than the significance level of α = 0.05.
Decision: Reject the Null Hypothesis H 0
Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

Because r is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of $r$ is significant or not . Compare r to the appropriate critical value in the table. If r is not between the positive and negative critical values, then the correlation coefficient is significant. If r is significant, then you may want to use the line for prediction.

For a given line of best fit, you computed that r = 0.6501 using n = 12 data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because r > the positive critical value.

For a given line of best fit, you compute that r = 0.5204 using n = 9 data points, and the critical value is 0.666. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because r < the positive critical value.

For a given line of best fit, you compute that r = –0.7204 using n = 8 data points, and the critical value is = 0.707. Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because r < the negative critical value.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Use the “95% Critical Value” table for r with df = n – 2 = 11 – 2 = 9.
The critical values are –0.602 and +0.602
Since 0.6631 > 0.602, r is significant.
Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score ( x ) and the final exam score ( y ) because the correlation coefficient is significantly different from zero.

r = –0.567 and the sample size, n , is 19. The df = n – 2 = 17. The critical value is –0.456. –0.567 < –0.456 so r is significant.
r = 0.708 and the sample size, n , is nine. The df = n – 2 = 7. The critical value is 0.666. 0.708 > 0.666 so r is significant.
r = 0.134 and the sample size, n , is 14. The df = 14 – 2 = 12. The critical value is 0.532. 0.134 is between –0.532 and 0.532 so r is not significant.
r = 0 and the sample size, n , is five. No matter what the dfs are, r = 0 is between the two critical values so r is not significant.

For a given line of best fit, you compute that r = 0 using n = 100 data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between x and y in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between x and y in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatterplot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

There is a linear relationship in the population that models the average value of y for varying values of x . In other words, the expected value of y for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
The y values for any particular x value are normally distributed about the line. This implies that there are more y values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of y values lie on the line.
The standard deviations of the population y values about the line are equal for each value of x . In other words, each of these normal distributions of y values has the same shape and spread about the line.
The residual errors are mutually independent (no pattern).
The data are produced from a well-designed, random sample or randomized experiment.

Chapter Review

Linear regression is a procedure for fitting a straight line of the form ŷ = a + bx to data. The conditions for regression are:

Linear In the population, there is a linear relationship that models the average value of y for different values of x .
Independent The residuals are assumed to be independent.
Normal The y values are distributed normally for any value of x .
Equal variance The standard deviation of the y values is equal for each x value.
Random The data are produced from a well-designed random sample or randomized experiment.

The slope b and intercept a of the least-squares line estimate the slope β and intercept α of the population (true) regression line. To estimate the population standard deviation of y , σ , use the standard deviation of the residuals, s . $s=\sqrt{\frac{SEE}{n-2}}$. The variable ρ (rho) is the population correlation coefficient. To test the null hypothesis H 0 : ρ = hypothesized value , use a linear regression t-test. The most common null hypothesis is H 0 : ρ = 0 which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

Formula Review

Least Squares Line or Line of Best Fit:

$\stackrel{^}{y}=a+bx$

a = y -intercept

Standard deviation of the residuals:

$s=\sqrt{\frac{SEE}{n-2}}.$

SSE = sum of squared errors

n = the number of data points

When testing the significance of the correlation coefficient, what is the null hypothesis?

When testing the significance of the correlation coefficient, what is the alternative hypothesis?

If the level of significance is 0.05 and the p -value is 0.04, what conclusion can you draw?

If the level of significance is 0.05 and the p -value is 0.06, what conclusion can you draw?

We do not reject the null hypothesis. There is not sufficient evidence to conclude that there is a significant linear relationship between x and y because the correlation coefficient is not significantly different from zero.

If there are 15 data points in a set of data, what is the number of degree of freedom?

JMP | Statistical Discovery.™ From SAS.

Statistics Knowledge Portal

A free online introduction to statistics

Correlation Coefficient

What is the correlation coefficient.

The correlation coefficient is the specific measure that quantifies the strength of the linear relationship between two variables in a correlation analysis. The coefficient is what we symbolize with the r in a correlation report.

How is the correlation coefficient used?

For two variables, the formula compares the distance of each datapoint from the variable mean and uses this to tell us how closely the relationship between the variables can be fit to an imaginary line drawn through the data. This is what we mean when we say that correlations look at linear relationships.

What are some limitations to consider?

Correlation only looks at the two variables at hand and won’t give insight into relationships beyond the bivariate data. This test won’t detect (and therefore will be skewed by) outliers in the data and can’t properly detect curvilinear relationships.

Correlation coefficient variants

This page focuses on the Pearson product-moment correlation. This is one of the most common types of correlation measures used in practice, but there are others. One closely related variant is the Spearman correlation, which is similar in usage but applicable to ranked data.

What do the values of the correlation coefficient mean?

See how to assess correlations using statistical software.

Download JMP to follow along using the sample data included with the software.
To see more JMP tutorials, visit the JMP Learning Library .

The correlation coefficient r is a unit-free value between -1 and 1. Statistical significance is indicated with a p-value. Therefore, correlations are typically written with two key numbers: r = and p = .

The closer r is to zero, the weaker the linear relationship.
Positive r values indicate a positive correlation, where the values of both variables tend to increase together.
Negative r values indicate a negative correlation, where the values of one variable tend to increase when the values of the other variable decrease.
The values 1 and -1 both represent "perfect" correlations, positive and negative respectively. Two perfectly correlated variables change together at a fixed rate. We say they have a linear relationship; when plotted on a scatterplot, all data points can be connected with a straight line.
The p-value helps us determine whether or not we can meaningfully conclude that the population correlation coefficient is different from zero, based on what we observe from the sample.

What is a p-value?

A p-value is a measure of probability used for hypothesis testing. The goal of hypothesis testing is to determine whether there is enough evidence to support a certain hypothesis about your data. Actually, we formulate two hypotheses: the null hypothesis and the alternative hypothesis. In the case of correlation analysis, the null hypothesis is typically that the observed relationship between the variables is the result of pure chance (i.e. the correlation coefficient is really zero — there is no linear relationship). The alternative hypothesis is that the correlation we’ve measured is legitimately present in our data (i.e. the correlation coefficient is different from zero).

The p-value is the probability of observing a non-zero correlation coefficient in our sample data when in fact the null hypothesis is true. A low p-value would lead you to reject the null hypothesis. A typical threshold for rejection of the null hypothesis is a p-value of 0.05. That is, if you have a p-value less than 0.05, you would reject the null hypothesis in favor of the alternative hypothesis—that the correlation coefficient is different from zero.

How do we actually calculate the correlation coefficient?

The sample correlation coefficient can be represented with a formula:

$$ r=\frac{\sum\left[\left(x_i-\overline{x}\right)\left(y_i-\overline{y}\right)\right]}{\sqrt{\mathrm{\Sigma}\left(x_i-\overline{x}\right)^2\ \ast\ \mathrm{\Sigma}(y_i\ -\overline{y})^2}} $$

View Annotated Formula

Let’s step through how to calculate the correlation coefficient using an example with a small set of simple numbers, so that it’s easy to follow the operations.

Let’s imagine that we’re interested in whether we can expect there to be more ice cream sales in our city on hotter days. Ice cream shops start to open in the spring; perhaps people buy more ice cream on days when it’s hot outside. On the other hand, perhaps people simply buy ice cream at a steady rate because they like it so much.

We start to answer this question by gathering data on average daily ice cream sales and the highest daily temperature. Ice Cream Sales and Temperature are therefore the two variables which we’ll use to calculate the correlation coefficient. Sometimes data like these are called bivariate data , because each observation (or point in time at which we’ve measured both sales and temperature) has two pieces of information that we can use to describe it. In other words, we’re asking whether Ice Cream Sales and Temperature seem to move together.

As before, a useful way to take a first look is with a scatterplot:

We can also look at these data in a table, which is handy for helping us follow the coefficient calculation for each datapoint. When talking about bivariate data, it’s typical to call one variable X and the other Y (these also help us orient ourselves on a visual plane, such as the axes of a plot). Let’s call Ice Cream Sales X , and Temperature Y .

Notice that each datapoint is paired . Remember, we are really looking at individual points in time, and each time has a value for both sales and temperature.

1. Start by finding the sample means

Now that we’re oriented to our data, we can start with two important subcalculations from the formula above: the sample mean , and the difference between each datapoint and this mean (in these steps, you can also see the initial building blocks of standard deviation ).

The sample means are represented with the symbols x̅ and y̅ , sometimes called “x bar” and “y bar.” The means for Ice Cream Sales ( x̅ ) and Temperature ( y̅ ) are easily calculated as follows:

$$ \overline{x} =\ [3\ +\ 6\ +\ 9] ÷ 3 = 6 $$

$$ \overline{y} =\ [70\ +\ 75\ +\ 80] ÷ 3 = 75 $$

2. Calculate the distance of each datapoint from its mean

With the mean in hand for each of our two variables, the next step is to subtract the mean of Ice Cream Sales (6) from each of our Sales data points ( x i in the formula), and the mean of Temperature (75) from each of our Temperature data points ( y i in the formula). Note that this operation sometimes results in a negative number or zero!

3. Complete the top of the coefficient equation

This piece of the equation is called the Sum of Products. A product is a number you get after multiplying, so this formula is just what it sounds like: the sum of numbers you multiply.

$$ \sum[(x_i-\overline{x})(y_i-\overline{y})] $$

We take the paired values from each row in the last two columns in the table above, multiply them (remember that multiplying two negative numbers makes a positive!), and sum those results:

$$ [(-3)(-5)] + [(0)(0)] + [(3)(5)] = 30 $$

How does the Sum of Products relate to the scatterplot?

The Sum of Products calculation and the location of the data points in our scatterplot are intrinsically related.

Notice that the Sum of Products is positive for our data. When the Sum of Products (the numerator of our correlation coefficient equation) is positive, the correlation coefficient r will be positive, since the denominator—a square root—will always be positive. We know that a positive correlation means that increases in one variable are associated with increases in the other (like our Ice Cream Sales and Temperature example), and on a scatterplot, the data points angle upwards from left to right. But how does the Sum of Products capture this?

The only way we will get a positive value for the Sum of Products is if the products we are summing tend to be positive.
The only way to get a positive value for each of the products is if both values are negative or both values are positive.
The only way to get a pair of two negative numbers is if both values are below their means (on the bottom left side of the scatter plot), and the only way to get a pair of two positive numbers is if both values are above their means (on the top right side of the scatter plot).

So, the Sum of Products tells us whether data tend to appear in the bottom left and top right of the scatter plot (a positive correlation), or alternatively, if the data tend to appear in the top left and bottom right of the scatter plot (a negative correlation).

4. Complete the bottom of the coefficient equation

The denominator of our correlation coefficient equation looks like this:

$$ \sqrt{\mathrm{\Sigma}{(x_i\ -\ \overline{x})}^2\ \ast\ \mathrm{\Sigma}(y_i\ -\overline{y})^2} $$

Let's tackle the expressions in this equation separately and drop in the numbers from our Ice Cream Sales example:

$$ \mathrm{\Sigma}{(x_i\ -\ \overline{x})}^2=-3^2+0^2+3^2=9+0+9=18 $$

$$ \mathrm{\Sigma}{(y_i\ -\ \overline{y})}^2=-5^2+0^2+5^2=25+0+25=50 $$

When we multiply the result of the two expressions together, we get:

$$ 18\times50\ =\ 900 $$

This brings the bottom of the equation to:

$$ \sqrt{900}=30 $$

5. Finish the calculation, and compare our result with the scatterplot

Here's our full correlation coefficient equation once again:

Let's pull in the numbers for the numerator and denominator that we calculated above:

$$ r=\frac{30}{30}=1 $$

A perfect correlation between ice cream sales and hot summer days! Of course, finding a perfect correlation is so unlikely in the real world that had we been working with real data, we’d assume we had done something wrong to obtain such a result.

But this result from the simplified data in our example should make intuitive sense based on simply looking at the data points. Let's look again at our scatterplot:

Now imagine drawing a line through that scatterplot. Would it look like a perfect linear fit?

A picture can be worth 1,000 correlation coefficients!

Scatterplots, and other data visualizations, are useful tools throughout the whole statistical process, not just before we perform our hypothesis tests.

In fact, it’s important to remember that relying exclusively on the correlation coefficient can be misleading—particularly in situations involving curvilinear relationships or extreme outliers. In the scatterplots below, we are reminded that a correlation coefficient of zero or near zero does not necessarily mean that there is no relationship between the variables; it simply means that there is no linear relationship.

Similarly, looking at a scatterplot can provide insights on how outliers—unusual observations in our data—can skew the correlation coefficient. Let’s look at an example with one extreme outlier. The correlation coefficient indicates that there is a relatively strong positive relationship between X and Y. But when the outlier is removed, the correlation coefficient is near zero.

Correlation and Regression with R

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Contributing Authors:
Learning Objectives
The Dataset

Correlation

Pearson correlation, spearman's rank correlation, some notes on correlation.

Simple Linear Regression
Introduction
Simple Linear Regression Model Fitting
Other Functions for Fitted Linear Model Objects
Multiple Linear Regression
Model Specification and Output
Model with Categorical Variables or Factors
Regression Diagnostics
Model Assumptions
Diagnostic Plots
More Diagnostics

Intro to R Contents

Common R Commands

Correlation is one of the most common statistics. Using one single value, it describes the "degree of relationship" between two variables. Correlation ranges from -1 to +1. Negative values of correlation indicate that as one variable increases the other variable decreases. Positive values of correlation indicate that as one variable increase the other variable increases as well. There are three options to calculate correlation in R, and we will introduce two of them below.

For a nice synopsis of correlation, see https://statistics.laerd.com/statistical-guides/pearson-correlation-coefficient-statistical-guide.php

The most commonly used type of correlation is Pearson correlation, named after Karl Pearson, introduced this statistic around the turn of the 20 th century. Pearson's r measures the linear relationship between two variables, say X and Y . A correlation of 1 indicates the data points perfectly lie on a line for which Y increases as X increases. A value of -1 also implies the data points lie on a line; however, Y decreases as X increases. The formula for r is

(in the same way that we distinguish between Ȳ and µ, similarly we distinguish r from ρ)

The Pearson correlation has two assumptions:

The two variables are normally distributed. We can test this assumption using
A statistical test (Shapiro-Wilk)
A histogram
The relationship between the two variables is linear. If this relationship is found to be curved, etc. we need to use another correlation test. We can test this assumption by examining the scatterplot between the two variables.

To calculate Pearson correlation, we can use the cor() function . The default method for cor() is the Pearson correlation. Getting a correlation is generally only half the story, and you may want to know if the relationship is statistically significantly different from 0.

H 0 : There is no correlation between the two variables: ρ = 0
H a : There is a nonzero correlation between the two variables: ρ ≠ 0

To assess statistical significance, you can use cor.test() function.

> cor(fat$age, fat$pctfat.brozek, method="pearson")

[1] 0.2891735

> cor.test(fat$age, fat$pctfat.brozek, method="pearson")

Pearson's product-moment correlation

data: fat$age and fat$pctfat.brozek

t = 4.7763, df = 250, p-value = 3.045e-06

alternative hypothesis: true correlation is not equal to 0

95 percent confidence interval:

0.1717375 0.3985061

sample estimates:

cor

When testing the null hypothesis that there is no correlation between age and Brozek percent body fat, we reject the null hypothesis (r = 0.289, t = 4.77, with 250 degrees of freedom, and a p-value = 3.045e-06). As age increases so does Brozek percent body fat. The 95% confidence interval for the correlation between age and Brozek percent body fat is (0.17, 0.40). Note that this 95% confidence interval does not contain 0, which is consistent with our decision to reject the null hypothesis.

Spearman's rank correlation is a nonparametric measure of the correlation that uses the rank of observations in its calculation, rather than the original numeric values. It measures the monotonic relationship between two variables X and Y. That is, if Y tends to increase as X increases, the Spearman correlation coefficient is positive. If Y tends to decrease as X increases, the Spearman correlation coefficient is negative. A value of zero indicates that there is no tendency for Y to either increase or decrease when X increases. The Spearman correlation measurement makes no assumptions about the distribution of the data.

The formula for Spearman's correlation ρ s is

where d i is the difference in the ranked observations from each group, ( x i – y i ), and n is the sample size. No need to memorize this formula!

> cor(fat$age,fat$pctfat.brozek, method="spearman")

[1] 0.2733830

> cor.test(fat$age,fat$pctfat.brozek, method="spearman")

Spearman's rank correlation rho

S = 1937979, p-value = 1.071e-05

alternative hypothesis: true rho is not equal to 0

rho

Thus we reject the null hypothesis that there is no (Spearman) correlation between age and Brozek percent fat (r = 0.27, p-value = 1.07e-05). As age increases so does percent body fat.

Correlation, useful though it is, is one of the most misused statistics in all of science. People always seem to want a simple number describing a relationship. Yet data very, very rarely obey this imperative. It is clear what a Pearson correlation of 1 or -1 means, but how do we interpret a correlation of 0.4? It is not so clear.

To see how the Pearson measure is dependent on the data distribution assumptions (in particular linearity), observe the following deterministic relationship: y = x 2 . Here the relationship between x and y isn't just "correlated," in the colloquial sense, it is totally deterministic ! If we generate data for this relationship, the Pearson correlation is 0!

> x<-seq(-10,10, 1)

> y<-x*x

> plot(x,y)

> cor(x,y)

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9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H 0 : μ __ 66
H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H 0 : μ __ 45
H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

H 0 : p __ 0.40
H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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How to Write a Hypothesis for Correlation

A hypothesis for correlation predicts a statistically significant relationship.

How to Calculate a P-Value

A hypothesis is a testable statement about how something works in the natural world. While some hypotheses predict a causal relationship between two variables, other hypotheses predict a correlation between them. According to the Research Methods Knowledge Base, a correlation is a single number that describes the relationship between two variables. If you do not predict a causal relationship or cannot measure one objectively, state clearly in your hypothesis that you are merely predicting a correlation.

Research the topic in depth before forming a hypothesis. Without adequate knowledge about the subject matter, you will not be able to decide whether to write a hypothesis for correlation or causation. Read the findings of similar experiments before writing your own hypothesis.

Identify the independent variable and dependent variable. Your hypothesis will be concerned with what happens to the dependent variable when a change is made in the independent variable. In a correlation, the two variables undergo changes at the same time in a significant number of cases. However, this does not mean that the change in the independent variable causes the change in the dependent variable.

Construct an experiment to test your hypothesis. In a correlative experiment, you must be able to measure the exact relationship between two variables. This means you will need to find out how often a change occurs in both variables in terms of a specific percentage.

Establish the requirements of the experiment with regard to statistical significance. Instruct readers exactly how often the variables must correlate to reach a high enough level of statistical significance. This number will vary considerably depending on the field. In a highly technical scientific study, for instance, the variables may need to correlate 98 percent of the time; but in a sociological study, 90 percent correlation may suffice. Look at other studies in your particular field to determine the requirements for statistical significance.

State the null hypothesis. The null hypothesis gives an exact value that implies there is no correlation between the two variables. If the results show a percentage equal to or lower than the value of the null hypothesis, then the variables are not proven to correlate.

Record and summarize the results of your experiment. State whether or not the experiment met the minimum requirements of your hypothesis in terms of both percentage and significance.

How to determine the sample size in a quantitative..., how to calculate a two-tailed test, how to interpret a student's t-test results, how to know if something is significant using spss, quantitative vs. qualitative data and laboratory testing, similarities of univariate & multivariate statistical..., what is the meaning of sample size, distinguishing between descriptive & causal studies, how to calculate cv values, how to determine your practice clep score, what are the different types of correlations, how to calculate p-hat, how to calculate percentage error, how to calculate percent relative range, how to calculate a sample size population, how to calculate bias, how to calculate the percentage of another number, how to find y value for the slope of a line, advantages & disadvantages of finding variance.

University of New England; Steps in Hypothesis Testing for Correlation; 2000
Research Methods Knowledge Base; Correlation; William M.K. Trochim; 2006
Science Buddies; Hypothesis

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6.3 - testing for partial correlation.

When discussing ordinary correlations we looked at tests for the null hypothesis that the ordinary correlation is equal to zero, against the alternative that it is not equal to zero. If that null hypothesis is rejected, then we look at confidence intervals for the ordinary correlation. Similar objectives can be considered for the partial correlation.

First, consider testing the null hypothesis that a partial correlation is equal to zero against the alternative that it is not equal to zero. This is expressed below:

$H_0\colon \rho_{jk\textbf{.x}}=0$ against $H_a\colon \rho_{jk\textbf{.x}}\ne 0$

Here we will use a test statistic that is similar to the one we used for an ordinary correlation. This test statistic is shown below:

$t = r_{jk\textbf{.x}}\sqrt{\frac{n-2-c}{1-r^2_{jk\textbf{.x}}}}$ $\dot{\sim}$ $t_{n-2-c}$

The only difference between this and the previous one is what appears in the numerator of the radical. Before we just took n - 2. Here we take n - 2 - c , where c is the number of variables upon which we are conditioning. In our Adult Intelligence data, we conditioned on two variables so c would be equal to 2 in this case.

Under the null hypothesis, this test statistic will be approximately t -distributed, also with n - 2 - c degrees of freedom.

We would reject $H_{o}\colon$ if the absolute value of the test statistic exceeded the critical value from the t -table evaluated at $\alpha$ over 2:

$|t| > t_{n-2-c, \alpha/2}$

Example 6-3: Wechsler Adult Intelligence Data Section

For the Wechsler Adult Intelligence Data, we found a partial correlation of 0.711879, which we enter into the expression for the test statistic as shown below:

$t = 0.711879 \sqrt{\dfrac{37-2-2}{1-0.711879^2}}=5.82$

The sample size is 37, along with the 2 variables upon which we are conditioning is also substituted in. Carry out the math and we get a test statistic of 5.82 as shown above.

Here we want to compare this value to a t -distribution with 33 degrees of freedom for an $\alpha$ = 0.01 level test. Therefore, we are going to look at the critical value for 0.005 in the table (because 33 does not appear to use the closest df that does not exceed 33 which is 30). In this case it is 2.75, meaning that $t _ { ( d f , 1 - \alpha / 2 ) } = t _ { ( 33,0.995 ) } $ is 2.75.

Because $5.82 > 2.75 = t _ { ( 33,0.995 ) }$, we can reject the null hypothesis, $H_{o}$ at the $\alpha = 0.01$ level and conclude that there is a significant partial correlation between these two variables. In particular, we would include that this partial correlation is positive indicating that even after taking into account Arithmetic and Picture Completion, there is a positive association between Information and Similarities.

Confidence Interval for the partial correlation, $\rho_{jk\textbf{.x}}$ Section

The procedure here is very similar to the procedure we used for ordinary correlation.

Compute Fisher's transformation of the partial correlation using the same formula as before.

$z_{jk} = \dfrac{1}{2}\log \left( \dfrac{1+r_{jk\textbf{.X}}}{1-r_{jk\textbf{.X}}}\right) $

In this case, for a large n , this Fisher transform variable will be possibly normally distributed. The mean is equal to the Fisher transform for the population value for this partial correlation, and the variance is equal to 1 over n-3-c .

$z_{jk}$ $\dot{\sim}$ $N \left( \dfrac{1}{2}\log \dfrac{1+\rho_{jk\textbf{.X}}}{1-\rho_{jk\textbf{.X}}}, \dfrac{1}{n-3-c}\right)$

Compute a $(1 - \alpha) × 100\%$ confidence interval for the Fisher transform correlation. This expression is shown below:

$ \dfrac{1}{2}\log \dfrac{1+\rho_{jk\textbf{.X}}}{1-\rho_{jk\textbf{.X}}}$

This yields the bounds $Z_{l}$ and $Z_{u}$ as before.

$\left(\underset{Z_l}{\underbrace{Z_{jk}-\dfrac{Z_{\alpha/2}}{\sqrt{n-3-c}}}}, \underset{Z_U}{\underbrace{Z_{jk}+\dfrac{Z_{\alpha/2}}{\sqrt{n-3-c}}}}\right)$

Back transform to obtain the desired confidence interval for the partial correlation - $\rho_{jk\textbf{.X}}$

$\left(\dfrac{e^{2Z_l}-1}{e^{2Z_l}+1}, \dfrac{e^{2Z_U}-1}{e^{2Z_U}+1}\right)$

Example 6-3: Wechsler Adult Intelligence Data (Steps Shown) Section

The confidence interval is calculated by substituting the results from the Wechsler Adult Intelligence Data into the appropriate steps below:

Step 1 : Compute the Fisher transform:

\begin{align} Z_{12} &= \dfrac{1}{2}\log \frac{1+r_{12.34}}{1-r_{12.34}}\\[5pt] &= \dfrac{1}{2} \log \frac{1+0.711879}{1-0.711879}\\[5pt] &= 0.89098 \end{align}

Step 2 : Compute the 95% confidence interval for $ \frac{1}{2}\log \frac{1+\rho_{12.34}}{1-\rho_{12.34}}$ :

\begin{align} Z_l &= Z_{12}-Z_{0.025}/\sqrt{n-3-c}\\[5pt] & = 0.89098 - \dfrac{1.96}{\sqrt{37-3-2}}\\[5pt] &= 0.5445 \end{align}

\begin{align} Z_U &= Z_{12}+Z_{0.025}/\sqrt{n-3-c}\\[5pt] &= 0.89098 + \dfrac{1.96}{\sqrt{37-3-2}} \\[5pt] &= 1.2375 \end{align}

Step 3 : Back-transform to obtain the 95% confidence interval for $\rho_{12.34}$ :

$\left(\dfrac{\exp\{2Z_l\}-1}{\exp\{2Z_l\}+1}, \dfrac{\exp\{2Z_U\}-1}{\exp\{2Z_U\}+1}\right)$

$\left(\dfrac{\exp\{2\times 0.5445\}-1}{\exp\{2\times 0.5445\}+1}, \dfrac{\exp\{2\times 1.2375\}-1}{\exp\{2\times 1.2375\}+1}\right)$

$(0.4964, 0.8447)$

Based on this result, we can conclude that we are 95% confident that the interval (0.4964, 0.8447) contains the partial correlation between Information and Similarities scores given scores on Arithmetic and Picture Completion.

Math Article

Null Hypothesis

In mathematics, Statistics deals with the study of research and surveys on the numerical data. For taking surveys, we have to define the hypothesis. Generally, there are two types of hypothesis. One is a null hypothesis, and another is an alternative hypothesis .

In probability and statistics, the null hypothesis is a comprehensive statement or default status that there is zero happening or nothing happening. For example, there is no connection among groups or no association between two measured events. It is generally assumed here that the hypothesis is true until any other proof has been brought into the light to deny the hypothesis. Let us learn more here with definition, symbol, principle, types and example, in this article.

Table of contents:

Comparison with Alternative Hypothesis

Null Hypothesis Definition

The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data. This hypothesis is either rejected or not rejected based on the viability of the given population or sample . In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0 .

Null Hypothesis Symbol

In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero), such that H 0 . It is pronounced as H-null or H-zero or H-nought. At the same time, the alternative hypothesis expresses the observations determined by the non-random cause. It is represented by H 1 or H a .

Null Hypothesis Principle

The principle followed for null hypothesis testing is, collecting the data and determining the chances of a given set of data during the study on some random sample, assuming that the null hypothesis is true. In case if the given data does not face the expected null hypothesis, then the outcome will be quite weaker, and they conclude by saying that the given set of data does not provide strong evidence against the null hypothesis because of insufficient evidence. Finally, the researchers tend to reject that.

Null Hypothesis Formula

Here, the hypothesis test formulas are given below for reference.

The formula for the null hypothesis is:

H 0 : p = p 0

The formula for the alternative hypothesis is:

H a = p >p 0 , < p 0 ≠ p 0

The formula for the test static is:

Remember that, p 0 is the null hypothesis and p – hat is the sample proportion.

Also, read:

Types of Null Hypothesis

There are different types of hypothesis. They are:

Simple Hypothesis

It completely specifies the population distribution. In this method, the sampling distribution is the function of the sample size.

Composite Hypothesis

The composite hypothesis is one that does not completely specify the population distribution.

Exact Hypothesis

Exact hypothesis defines the exact value of the parameter. For example μ= 50

Inexact Hypothesis

This type of hypothesis does not define the exact value of the parameter. But it denotes a specific range or interval. For example 45< μ <60

Null Hypothesis Rejection

Sometimes the null hypothesis is rejected too. If this hypothesis is rejected means, that research could be invalid. Many researchers will neglect this hypothesis as it is merely opposite to the alternate hypothesis. It is a better practice to create a hypothesis and test it. The goal of researchers is not to reject the hypothesis. But it is evident that a perfect statistical model is always associated with the failure to reject the null hypothesis.

How do you Find the Null Hypothesis?

The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don’t have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent).

When is Null Hypothesis Rejected?

The null hypothesis is rejected using the P-value approach. If the P-value is less than or equal to the α, there should be a rejection of the null hypothesis in favour of the alternate hypothesis. In case, if P-value is greater than α, the null hypothesis is not rejected.

Null Hypothesis and Alternative Hypothesis

Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.

Null Hypothesis Examples

Here, some of the examples of the null hypothesis are given below. Go through the below ones to understand the concept of the null hypothesis in a better way.

If a medicine reduces the risk of cardiac stroke, then the null hypothesis should be “the medicine does not reduce the chance of cardiac stroke”. This testing can be performed by the administration of a drug to a certain group of people in a controlled way. If the survey shows that there is a significant change in the people, then the hypothesis is rejected.

Few more examples are:

1). Are there is 100% chance of getting affected by dengue?

Ans: There could be chances of getting affected by dengue but not 100%.

2). Do teenagers are using mobile phones more than grown-ups to access the internet?

Ans: Age has no limit on using mobile phones to access the internet.

3). Does having apple daily will not cause fever?

Ans: Having apple daily does not assure of not having fever, but increases the immunity to fight against such diseases.

4). Do the children more good in doing mathematical calculations than grown-ups?

Ans: Age has no effect on Mathematical skills.

In many common applications, the choice of the null hypothesis is not automated, but the testing and calculations may be automated. Also, the choice of the null hypothesis is completely based on previous experiences and inconsistent advice. The choice can be more complicated and based on the variety of applications and the diversity of the objectives.

The main limitation for the choice of the null hypothesis is that the hypothesis suggested by the data is based on the reasoning which proves nothing. It means that if some hypothesis provides a summary of the data set, then there would be no value in the testing of the hypothesis on the particular set of data.

Frequently Asked Questions on Null Hypothesis

What is meant by the null hypothesis.

In Statistics, a null hypothesis is a type of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.

What are the benefits of hypothesis testing?

Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.

When a null hypothesis is accepted and rejected?

The null hypothesis is either accepted or rejected in terms of the given data. If P-value is less than α, then the null hypothesis is rejected in favor of the alternative hypothesis, and if the P-value is greater than α, then the null hypothesis is accepted in favor of the alternative hypothesis.

Why is the null hypothesis important?

The importance of the null hypothesis is that it provides an approximate description of the phenomena of the given data. It allows the investigators to directly test the relational statement in a research study.

How to accept or reject the null hypothesis in the chi-square test?

If the result of the chi-square test is bigger than the critical value in the table, then the data does not fit the model, which represents the rejection of the null hypothesis.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

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11.2: Correlation Hypothesis Test
The p-value is calculated using a t -distribution with n − 2 degrees of freedom. The formula for the test statistic is t = r√n − 2 √1 − r2. The value of the test statistic, t, is shown in the computer or calculator output along with the p-value. The test statistic t has the same sign as the correlation coefficient r.
1.9
In general, a researcher should use the hypothesis test for the population correlation \ (\rho\) to learn of a linear association between two variables, when it isn't obvious which variable should be regarded as the response. Let's clarify this point with examples of two different research questions. Consider evaluating whether or not a linear ...
12.4 Testing the Significance of the Correlation Coefficient
PERFORMING THE HYPOTHESIS TEST. Null Hypothesis: H 0: ρ = 0 Alternate Hypothesis: H a: ρ ≠ 0 WHAT THE HYPOTHESES MEAN IN WORDS: Null Hypothesis H 0: The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship (correlation) between x and y in the population.; Alternate Hypothesis H a: The population correlation coefficient ...
Pearson Correlation Coefficient (r)
Example: Deciding whether to reject the null hypothesis For the correlation between weight and height in a sample of 10 newborns, the t value is less than the critical value of t. Therefore, we don't reject the null hypothesis that the Pearson correlation coefficient of the population (ρ) is 0.
Null Hypothesis: Definition, Rejecting & Examples
Null Hypothesis H 0: The correlation in the population is zero: ρ = 0. Alternative Hypothesis H A: The correlation in the population is not zero: ρ ≠ 0. For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.
13.2 Testing the Significance of the Correlation Coefficient
Null Hypothesis: H 0: ρ = 0; Alternate ... Alternate Hypothesis H a: The population correlation coefficient is significantly different from zero. There is a significant linear relationship (correlation) between X 1 and X 2 in the population. Drawing a Conclusion There are two methods of making the decision concerning the hypothesis. The test ...
9.4.1
The test statistic is: t ∗ = r n − 2 1 − r 2 = ( 0.711) 28 − 2 1 − 0.711 2 = 5.1556. Next, we need to find the p-value. The p-value for the two-sided test is: p-value = 2 P ( T > 5.1556) < 0.0001. Therefore, for any reasonable α level, we can reject the hypothesis that the population correlation coefficient is 0 and conclude that it ...
Chapter 12.5: Testing the Significance of the Correlation Coefficient
The variable ρ (rho) is the population correlation coefficient. To test the null hypothesis H 0: ρ = hypothesized value, use a linear regression t-test. The most common null hypothesis is H 0: ρ = 0 which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can ...
Null & Alternative Hypotheses
Null hypothesis (H 0): There's no effect in the population. Alternative hypothesis (H a or H 1): There's an effect in the population. ... There is a correlation between independent variable and dependent variable in the population; ρ ≠ 0. Simple linear regression:
Testing the Significance of the Correlation Coefficient
The variable ρ (rho) is the population correlation coefficient. To test the null hypothesis H 0: ρ = hypothesized value, use a linear regression t-test. The most common null hypothesis is H 0: ρ = 0 which indicates there is no linear relationship between x and y in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can ...
Interpreting Correlation Coefficients
Null hypothesis: There is no linear relationship between the two variables. ... It's testing the null hypothesis that the correlation equals zero. Because your p-value is greater than any reasonable significance level, you fail to reject the null. Your data provide insufficient evidence to conclude that the correlation doesn't equal zero ...
5.3
5.3 - Inferences for Correlations. Let us consider testing the null hypothesis that there is zero correlation between two variables X j and X k. Mathematically we write this as shown below: H 0: ρ j k = 0 against H a: ρ j k ≠ 0. Recall that the correlation is estimated by sample correlation r j k given in the expression below: r j k = s j k ...
Correlation Coefficient
The alternative hypothesis is that the correlation we've measured is legitimately present in our data (i.e. the correlation coefficient is different from zero). The p-value is the probability of observing a non-zero correlation coefficient in our sample data when in fact the null hypothesis is true. A low p-value would lead you to reject the ...
Correlation
When testing the null hypothesis that there is no correlation between age and Brozek percent body fat, we reject the null hypothesis (r = 0.289, t = 4.77, with 250 degrees of freedom, and a p-value = 3.045e-06). As age increases so does Brozek percent body fat. The 95% confidence interval for the correlation between age and Brozek percent body ...
Everything you need to know about interpreting correlations
The null hypothesis is the hypothesis that we are trying to provide evidence against, in our case, we try to provide evidence againt the hypothesis that there is not a significant linear correlation between x and y in the population (i.e. ρ = 0) Null hypothesis Ho: ρ = 0; Alternative hypothesis Ha: ρ ≠ 0; Step 2: T-test
9.1 Null and Alternative Hypotheses
The actual test begins by considering two hypotheses.They are called the null hypothesis and the alternative hypothesis.These hypotheses contain opposing viewpoints. H 0, the —null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.
How to Write a Hypothesis for Correlation
A hypothesis is a testable statement about how something works in the natural world. While some hypotheses predict a causal relationship between two variables, other hypotheses predict a correlation between them. According to the Research Methods Knowledge Base, a correlation is a single number that describes the relationship between two variables.
6.3
First, consider testing the null hypothesis that a partial correlation is equal to zero against the alternative that it is not equal to zero. This is expressed below: H 0: ρ j k .x = 0 against H a: ρ j k .x ≠ 0. Here we will use a test statistic that is similar to the one we used for an ordinary correlation. This test statistic is shown below:
Null Hypothesis
The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don't have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent). ...

11.2: Correlation Hypothesis Test

PERFORMING THE HYPOTHESIS TEST

METHOD 1: Using a \(p\text{-value}\) to make a decision

METHOD 2: Using a table of Critical Values to make a decision

Example \(\PageIndex{1}\)

Exercise \(\PageIndex{1}\)

Example \(\PageIndex{2}\)

Exercise \(\PageIndex{2}\)

Example \(\PageIndex{3}\)

Exercise \(\PageIndex{3}\)

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Example \(\PageIndex{4}\)

Exercise \(\PageIndex{4}\)

Assumptions in Testing the Significance of the Correlation Coefficient

Formula Review

Chapter 12.5: Testing the Significance of the Correlation Coefficient

PERFORMING THE HYPOTHESIS TEST

METHOD 1: Using a p -value to make a decision

METHOD 2: Using a table of Critical Values to make a decision

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Assumptions in Testing the Significance of the Correlation Coefficient

Chapter Review

Formula Review

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Testing the Significance of the Correlation Coefficient

PERFORMING THE HYPOTHESIS TEST

METHOD 1: Using a p -value to make a decision

METHOD 2: Using a table of Critical Values to make a decision

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Assumptions in Testing the Significance of the Correlation Coefficient

Chapter Review

Formula Review

Statistics Knowledge Portal

Correlation Coefficient

How is the correlation coefficient used?

What are some limitations to consider?

Correlation coefficient variants

What do the values of the correlation coefficient mean?

How do we actually calculate the correlation coefficient?

1. Start by finding the sample means

2. Calculate the distance of each datapoint from its mean

3. Complete the top of the coefficient equation

4. Complete the bottom of the coefficient equation

5. Finish the calculation, and compare our result with the scatterplot

A picture can be worth 1,000 correlation coefficients!

Correlation

9.1 Null and Alternative Hypotheses

Example 9.1

Example 9.2

Example 9.3

Example 9.4

Collaborative Exercise

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How to Write a Hypothesis for Correlation

How to Calculate a P-Value

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Example 6-3: Wechsler Adult Intelligence Data Section

Confidence Interval for the partial correlation, \(\rho_{jk\textbf{.x}}\) Section

Example 6-3: Wechsler Adult Intelligence Data (Steps Shown) Section

Null Hypothesis

Null Hypothesis Definition

Null Hypothesis Symbol

Null Hypothesis Principle

Null Hypothesis Formula

Types of Null Hypothesis

Null Hypothesis Rejection

How do you Find the Null Hypothesis?

When is Null Hypothesis Rejected?

Null Hypothesis and Alternative Hypothesis

Null Hypothesis Examples

Frequently Asked Questions on Null Hypothesis

What are the benefits of hypothesis testing?

When a null hypothesis is accepted and rejected?

Why is the null hypothesis important?

How to accept or reject the null hypothesis in the chi-square test?

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