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8.6: Hypothesis Test of a Single Population Mean with Examples

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Steps for performing Hypothesis Test of a Single Population Mean

Step 1: State your hypotheses about the population mean. Step 2: Summarize the data. State a significance level. State and check conditions required for the procedure

• Find or identify the sample size, n, the sample mean, \(\bar{x}\) and the sample standard deviation, s .

The sampling distribution for the one-mean test statistic is, approximately, T- distribution if the following conditions are met

• Sample is random with independent observations .
• Sample is large. The population must be Normal or the sample size must be at least 30.

Step 3: Perform the procedure based on the assumption that \(H_{0}\) is true

• Find the Estimated Standard Error: \(SE=\frac{s}{\sqrt{n}}\).
• Compute the observed value of the test statistic: \(T_{obs}=\frac{\bar{x}-\mu_{0}}{SE}\).
• Check the type of the test (right-, left-, or two-tailed)
• Find the p-value in order to measure your level of surprise.

Step 4: Make a decision about \(H_{0}\) and \(H_{a}\)

• Do you reject or not reject your null hypothesis?

Step 5: Make a conclusion

• What does this mean in the context of the data?

The following examples illustrate a left-, right-, and two-tailed test.

Example \(\pageindex{1}\).

\(H_{0}: \mu = 5, H_{a}: \mu < 5\)

Test of a single population mean. \(H_{a}\) tells you the test is left-tailed. The picture of the \(p\)-value is as follows:

Exercise \(\PageIndex{1}\)

\(H_{0}: \mu = 10, H_{a}: \mu < 10\)

Assume the \(p\)-value is 0.0935. What type of test is this? Draw the picture of the \(p\)-value.

left-tailed test

Example \(\PageIndex{2}\)

\(H_{0}: \mu \leq 0.2, H_{a}: \mu > 0.2\)

This is a test of a single population proportion. \(H_{a}\) tells you the test is right-tailed . The picture of the p -value is as follows:

Exercise \(\PageIndex{2}\)

\(H_{0}: \mu \leq 1, H_{a}: \mu > 1\)

Assume the \(p\)-value is 0.1243. What type of test is this? Draw the picture of the \(p\)-value.

right-tailed test

Example \(\PageIndex{3}\)

\(H_{0}: \mu = 50, H_{a}: \mu \neq 50\)

This is a test of a single population mean. \(H_{a}\) tells you the test is two-tailed . The picture of the \(p\)-value is as follows.

Exercise \(\PageIndex{3}\)

\(H_{0}: \mu = 0.5, H_{a}: \mu \neq 0.5\)

Assume the p -value is 0.2564. What type of test is this? Draw the picture of the \(p\)-value.

two-tailed test

Full Hypothesis Test Examples

Example \(\pageindex{4}\).

Statistics students believe that the mean score on the first statistics test is 65. A statistics instructor thinks the mean score is higher than 65. He samples ten statistics students and obtains the scores 65 65 70 67 66 63 63 68 72 71. He performs a hypothesis test using a 5% level of significance. The data are assumed to be from a normal distribution.

Set up the hypothesis test:

A 5% level of significance means that \(\alpha = 0.05\). This is a test of a single population mean .

\(H_{0}: \mu = 65  H_{a}: \mu > 65\)

Since the instructor thinks the average score is higher, use a "\(>\)". The "\(>\)" means the test is right-tailed.

Determine the distribution needed:

Random variable: \(\bar{X} =\) average score on the first statistics test.

Distribution for the test: If you read the problem carefully, you will notice that there is no population standard deviation given . You are only given \(n = 10\) sample data values. Notice also that the data come from a normal distribution. This means that the distribution for the test is a student's \(t\).

Use \(t_{df}\). Therefore, the distribution for the test is \(t_{9}\) where \(n = 10\) and \(df = 10 - 1 = 9\).

The sample mean and sample standard deviation are calculated as 67 and 3.1972 from the data.

Calculate the \(p\)-value using the Student's \(t\)-distribution:

\[t_{obs} = \dfrac{\bar{x}-\mu_{\bar{x}}}{\left(\dfrac{s}{\sqrt{n}}\right)}=\dfrac{67-65}{\left(\dfrac{3.1972}{\sqrt{10}}\right)}\]

Use the T-table or Excel's t_dist() function to find p-value:

\(p\text{-value} = P(\bar{x} > 67) =P(T >1.9782 )= 1-0.9604=0.0396\)

Interpretation of the p -value: If the null hypothesis is true, then there is a 0.0396 probability (3.96%) that the sample mean is 65 or more.

Compare \(\alpha\) and the \(p-\text{value}\):

Since \(α = 0.05\) and \(p\text{-value} = 0.0396\). \(\alpha > p\text{-value}\).

Make a decision: Since \(\alpha > p\text{-value}\), reject \(H_{0}\).

This means you reject \(\mu = 65\). In other words, you believe the average test score is more than 65.

Conclusion: At a 5% level of significance, the sample data show sufficient evidence that the mean (average) test score is more than 65, just as the math instructor thinks.

The \(p\text{-value}\) can easily be calculated.

Put the data into a list. Press STAT and arrow over to TESTS . Press 2:T-Test . Arrow over to Data and press ENTER . Arrow down and enter 65 for \(\mu_{0}\), the name of the list where you put the data, and 1 for Freq: . Arrow down to \(\mu\): and arrow over to \(> \mu_{0}\). Press ENTER . Arrow down to Calculate and press ENTER . The calculator not only calculates the \(p\text{-value}\) (p = 0.0396) but it also calculates the test statistic ( t -score) for the sample mean, the sample mean, and the sample standard deviation. \(\mu > 65\) is the alternative hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate ). Press ENTER . A shaded graph appears with \(t = 1.9781\) (test statistic) and \(p = 0.0396\) (\(p\text{-value}\)). Make sure when you use Draw that no other equations are highlighted in \(Y =\) and the plots are turned off.

Exercise \(\PageIndex{4}\)

It is believed that a stock price for a particular company will grow at a rate of $5 per week with a standard deviation of $1. An investor believes the stock won’t grow as quickly. The changes in stock price is recorded for ten weeks and are as follows: $4, $3, $2, $3, $1, $7, $2, $1, $1, $2. Perform a hypothesis test using a 5% level of significance. State the null and alternative hypotheses, find the p -value, state your conclusion, and identify the Type I and Type II errors.

• \(H_{0}: \mu = 5\)
• \(H_{a}: \mu < 5\)
• \(p = 0.0082\)

Because \(p < \alpha\), we reject the null hypothesis. There is sufficient evidence to suggest that the stock price of the company grows at a rate less than $5 a week.

• Type I Error: To conclude that the stock price is growing slower than $5 a week when, in fact, the stock price is growing at $5 a week (reject the null hypothesis when the null hypothesis is true).
• Type II Error: To conclude that the stock price is growing at a rate of $5 a week when, in fact, the stock price is growing slower than $5 a week (do not reject the null hypothesis when the null hypothesis is false).

Example \(\PageIndex{5}\)

The National Institute of Standards and Technology provides exact data on conductivity properties of materials. Following are conductivity measurements for 11 randomly selected pieces of a particular type of glass.

1.11; 1.07; 1.11; 1.07; 1.12; 1.08; .98; .98 1.02; .95; .95

Is there convincing evidence that the average conductivity of this type of glass is greater than one? Use a significance level of 0.05. Assume the population is normal.

Let’s follow a four-step process to answer this statistical question.

• \(H_{0}: \mu \leq 1\)
• \(H_{a}: \mu > 1\)
• Plan : We are testing a sample mean without a known population standard deviation. Therefore, we need to use a Student's-t distribution. Assume the underlying population is normal.
• Do the calculations : \(p\text{-value} ( = 0.036)\)

4. State the Conclusions : Since the \(p\text{-value} (= 0.036)\) is less than our alpha value, we will reject the null hypothesis. It is reasonable to state that the data supports the claim that the average conductivity level is greater than one.

The hypothesis test itself has an established process. This can be summarized as follows:

• Determine \(H_{0}\) and \(H_{a}\). Remember, they are contradictory.
• Determine the random variable.
• Determine the distribution for the test.
• Draw a graph, calculate the test statistic, and use the test statistic to calculate the \(p\text{-value}\). (A t -score is an example of test statistics.)
• Compare the preconceived α with the p -value, make a decision (reject or do not reject H 0 ), and write a clear conclusion using English sentences.

Notice that in performing the hypothesis test, you use \(\alpha\) and not \(\beta\). \(\beta\) is needed to help determine the sample size of the data that is used in calculating the \(p\text{-value}\). Remember that the quantity \(1 – \beta\) is called the Power of the Test . A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping α the same.If the power is low, the null hypothesis might not be rejected when it should be.

• Data from Amit Schitai. Director of Instructional Technology and Distance Learning. LBCC.
• Data from Bloomberg Businessweek . Available online at www.businessweek.com/news/2011- 09-15/nyc-smoking-rate-falls-to-record-low-of-14-bloomberg-says.html.
• Data from energy.gov. Available online at http://energy.gov (accessed June 27. 2013).
• Data from Gallup®. Available online at www.gallup.com (accessed June 27, 2013).
• Data from Growing by Degrees by Allen and Seaman.
• Data from La Leche League International. Available online at www.lalecheleague.org/Law/BAFeb01.html.
• Data from the American Automobile Association. Available online at www.aaa.com (accessed June 27, 2013).
• Data from the American Library Association. Available online at www.ala.org (accessed June 27, 2013).
• Data from the Bureau of Labor Statistics. Available online at http://www.bls.gov/oes/current/oes291111.htm .
• Data from the Centers for Disease Control and Prevention. Available online at www.cdc.gov (accessed June 27, 2013)
• Data from the U.S. Census Bureau, available online at quickfacts.census.gov/qfd/states/00000.html (accessed June 27, 2013).
• Data from the United States Census Bureau. Available online at www.census.gov/hhes/socdemo/language/.
• Data from Toastmasters International. Available online at http://toastmasters.org/artisan/deta...eID=429&Page=1 .
• Data from Weather Underground. Available online at www.wunderground.com (accessed June 27, 2013).
• Federal Bureau of Investigations. “Uniform Crime Reports and Index of Crime in Daviess in the State of Kentucky enforced by Daviess County from 1985 to 2005.” Available online at http://www.disastercenter.com/kentucky/crime/3868.htm (accessed June 27, 2013).
• “Foothill-De Anza Community College District.” De Anza College, Winter 2006. Available online at research.fhda.edu/factbook/DA...t_da_2006w.pdf.
• Johansen, C., J. Boice, Jr., J. McLaughlin, J. Olsen. “Cellular Telephones and Cancer—a Nationwide Cohort Study in Denmark.” Institute of Cancer Epidemiology and the Danish Cancer Society, 93(3):203-7. Available online at http://www.ncbi.nlm.nih.gov/pubmed/11158188 (accessed June 27, 2013).
• Rape, Abuse & Incest National Network. “How often does sexual assault occur?” RAINN, 2009. Available online at www.rainn.org/get-information...sexual-assault (accessed June 27, 2013).

Hypothesis Testing Framework

Now that we've seen an example and explored some of the themes for hypothesis testing, let's specify the procedure that we will follow.

Hypothesis Testing Steps

The formal framework and steps for hypothesis testing are as follows:

• Identify and define the parameter of interest
• Define the competing hypotheses to test
• Set the evidence threshold, formally called the significance level
• Generate or use theory to specify the sampling distribution and check conditions
• Calculate the test statistic and p-value
• Evaluate your results and write a conclusion in the context of the problem.

We'll discuss each of these steps below.

Identify Parameter of Interest

First, I like to specify and define the parameter of interest. What is the population that we are interested in? What characteristic are we measuring?

By defining our population of interest, we can confirm that we are truly using sample data. If we find that we actually have population data, our inference procedures are not needed. We could proceed by summarizing our population data.

By identifying and defining the parameter of interest, we can confirm that we use appropriate methods to summarize our variable of interest. We can also focus on the specific process needed for our parameter of interest.

In our example from the last page, the parameter of interest would be the population mean time that a host has been on Airbnb for the population of all Chicago listings on Airbnb in March 2023. We could represent this parameter with the symbol $\mu$. It is best practice to fully define $\mu$ both with words and symbol.

Define the Hypotheses

For hypothesis testing, we need to decide between two competing theories. These theories must be statements about the parameter. Although we won't have the population data to definitively select the correct theory, we will use our sample data to determine how reasonable our "skeptic's theory" is.

The first hypothesis is called the null hypothesis, $H_0$. This can be thought of as the "status quo", the "skeptic's theory", or that nothing is happening.

Examples of null hypotheses include that the population proportion is equal to 0.5 ($p = 0.5$), the population median is equal to 12 ($M = 12$), or the population mean is equal to 14.5 ($\mu = 14.5$).

The second hypothesis is called the alternative hypothesis, $H_a$ or $H_1$. This can be thought of as the "researcher's hypothesis" or that something is happening. This is what we'd like to convince the skeptic to believe. In most cases, the desired outcome of the researcher is to conclude that the alternative hypothesis is reasonable to use moving forward.

Examples of alternative hypotheses include that the population proportion is greater than 0.5 ($p > 0.5$), the population median is less than 12 ($M < 12$), or the population mean is not equal to 14.5 ($\mu \neq 14.5$).

There are a few requirements for the hypotheses:

• the hypotheses must be about the same population parameter,
• the hypotheses must have the same null value (provided number to compare to),
• the null hypothesis must have the equality (the equals sign must be in the null hypothesis),
• the alternative hypothesis must not have the equality (the equals sign cannot be in the alternative hypothesis),
• there must be no overlap between the null and alternative hypothesis.

You may have previously seen null hypotheses that include more than an equality (e.g. $p \le 0.5$). As long as there is an equality in the null hypothesis, this is allowed. For our purposes, we will simplify this statement to ($p = 0.5$).

To summarize from above, possible hypotheses statements are:

$H_0: p = 0.5$ vs. $H_a: p > 0.5$

$H_0: M = 12$ vs. $H_a: M < 12$

$H_0: \mu = 14.5$ vs. $H_a: \mu \neq 14.5$

In our second example about Airbnb hosts, our hypotheses would be:

$H_0: \mu = 2100$ vs. $H_a: \mu > 2100$.

Set Threshold (Significance Level)

There is one more step to complete before looking at the data. This is to set the threshold needed to convince the skeptic. This threshold is defined as an $\alpha$ significance level. We'll define exactly what the $\alpha$ significance level means later. For now, smaller $\alpha$s correspond to more evidence being required to convince the skeptic.

A few common $\alpha$ levels include 0.1, 0.05, and 0.01.

For our Airbnb hosts example, we'll set the threshold as 0.02.

Determine the Sampling Distribution of the Sample Statistic

The first step (as outlined above) is the identify the parameter of interest. What is the best estimate of the parameter of interest? Typically, it will be the sample statistic that corresponds to the parameter. This sample statistic, along with other features of the distribution will prove especially helpful as we continue the hypothesis testing procedure.

However, we do have a decision at this step. We can choose to use simulations with a resampling approach or we can choose to rely on theory if we are using proportions or means. We then also need to confirm that our results and conclusions will be valid based on the available data.

Required Condition

The one required assumption, regardless of approach (resampling or theory), is that the sample is random and representative of the population of interest. In other words, we need our sample to be a reasonable sample of data from the population.

Using Simulations and Resampling

If we'd like to use a resampling approach, we have no (or minimal) additional assumptions to check. This is because we are relying on the available data instead of assumptions.

We do need to adjust our data to be consistent with the null hypothesis (or skeptic's claim). We can then rely on our resampling approach to estimate a plausible sampling distribution for our sample statistic.

Recall that we took this approach on the last page. Before simulating our estimated sampling distribution, we adjusted the mean of the data so that it matched with our skeptic's claim, shown in the code below.

We'll see a few more examples on the next page.

Using Theory

On the other hand, we could rely on theory in order to estimate the sampling distribution of our desired statistic. Recall that we had a few different options to rely on:

• the CLT for the sampling distribution of a sample mean
• the binomial distribution for the sampling distribution of a proportion (or count)
• the Normal approximation of a binomial distribution (using the CLT) for the sampling distribution of a proportion

If relying on the CLT to specify the underlying sampling distribution, you also need to confirm:

• having a random sample and
• having a sample size that is less than 10% of the population size if the sampling is done without replacement
• having a Normally distributed population for a quantitative variable OR
• having a large enough sample size (usually at least 25) for a quantitative variable
• having a large enough sample size for a categorical variable (defined by $np$ and $n(1-p)$ being at least 10)

If relying on the binomial distribution to specify the underlying sampling distribution, you need to confirm:

• having a set number of trials, $n$
• having the same probability of success, $p$ for each observation

After determining the appropriate theory to use, we should check our conditions and then specify the sampling distribution for our statistic.

For the Airbnb hosts example, we have what we've assumed to be a random sample. It is not taken with replacement, so we also need to assume that our sample size (700) is less than 10% of our population size. In other words, we need to assume that the population of Chicago Airbnbs in March 2023 was at least 7000. Since we do have our (presumed) population data available, we can confirm that there were at least 7000 Chicago Airbnbs in the population in 2023.

Additionally, we can confirm that normality of the sampling distribution applies for the CLT to apply. Our sample size is more than 25 and the parameter of interest is a mean, so this meets our necessary criteria for the normality condition to be valid.

With the conditions now met, we can estimate our sampling distribution. From the CLT, we know that the distribution for the sample mean should be $\bar{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}})$.

Now, we face our next challenge -- what to plug in as the mean and standard error for this distribution. Since we are adopting the skeptic's point of view for the purpose of this approach, we can plug in the value of $\mu_0 = 2100$. We also know that the sample size $n$ is 700. But what should we plug in for the population standard deviation $\sigma$?

When we don't know the value of a parameter, we will generally plug in our best estimate for the parameter. In this case, that corresponds to plugging in $\hat{\sigma}$, or our sample standard deviation.

Now, our estimated sampling distribution based on the CLT is: $\bar{X} \sim N(2100, 41.4045)$.

If we compare to our corresponding skeptic's sampling distribution on the last page, we can confirm that the theoretical sampling distribution is similar to the simulated sampling distribution based on resampling.

Assumptions not met

What do we do if the necessary conditions aren't met for the sampling distribution? Because the simulation-based resampling approach has minimal assumptions, we should be able to use this approach to produce valid results as long as the provided data is representative of the population.

The theory-based approach has more conditions, and we may not be able to meet all of the necessary conditions. For example, if our parameter is something other than a mean or proportion, we may not have appropriate theory. Additionally, we may not have a large enough sample size.

• First, we could consider changing approaches to the simulation-based one.
• Second, we might look at how we could meet the necessary conditions better. In some cases, we may be able to redefine groups or make adjustments so that the setup of the test is closer to what is needed.
• As a last resort, we may be able to continue following the hypothesis testing steps. In this case, your calculations may not be valid or exact; however, you might be able to use them as an estimate or an approximation. It would be crucial to specify the violation and approximation in any conclusions or discussion of the test.

Calculate the evidence with statistics and p-values

Now, it's time to calculate how much evidence the sample contains to convince the skeptic to change their mind. As we saw above, we can convince the skeptic to change their mind by demonstrating that our sample is unlikely to occur if their theory is correct.

How do we do this? We do this by calculating a probability associated with our observed value for the statistic.

For example, for our situation, we want to convince the skeptic that the population mean is actually greater than 2100 days. We do that by calculating the probability that a sample mean would be as large or larger than what we observed in our actual sample, which was 2188 days. Why do we need the larger portion? We use the larger portion because a sample mean of 2200 days also provides evidence that the population mean is larger than 2100 days; it isn't limited to exactly what we observed in our sample. We call this specific probability the p-value.

That is, the p-value is the probability of observing a test statistic as extreme or more extreme (as determined by the alternative hypothesis), assuming the null hypothesis is true.

Our observed p-value for the Airbnb host example demonstrates that the probability of getting a sample mean host time of 2188 days (the value from our sample) or more is 1.46%, assuming that the true population mean is 2100 days.

Test statistic

Notice that the formal definition of a p-value mentions a test statistic . In most cases, this word can be replaced with "statistic" or "sample" for an equivalent statement.

Oftentimes, we'll see that our sample statistic can be used directly as the test statistic, as it was above. We could equivalently adjust our statistic to calculate a test statistic. This test statistic is often calculated as:

$\text{test statistic} = \frac{\text{estimate} - \text{hypothesized value}}{\text{standard error of estimate}}$

P-value Calculation Options

Note also that the p-value definition includes a probability associated with a test statistic being as extreme or more extreme (as determined by the alternative hypothesis . How do we determine the area that we consider when calculating the probability. This decision is determined by the inequality in the alternative hypothesis.

For example, when we were trying to convince the skeptic that the population mean is greater than 2100 days, we only considered those sample means that we at least as large as what we observed -- 2188 days or more.

If instead we were trying to convince the skeptic that the population mean is less than 2100 days ($H_a: \mu < 2100$), we would consider all sample means that were at most what we observed - 2188 days or less. In this case, our p-value would be quite large; it would be around 99.5%. This large p-value demonstrates that our sample does not support the alternative hypothesis. In fact, our sample would encourage us to choose the null hypothesis instead of the alternative hypothesis of $\mu < 2100$, as our sample directly contradicts the statement in the alternative hypothesis.

If we wanted to convince the skeptic that they were wrong and that the population mean is anything other than 2100 days ($H_a: \mu \neq 2100$), then we would want to calculate the probability that a sample mean is at least 88 days away from 2100 days. That is, we would calculate the probability corresponding to 2188 days or more or 2012 days or less. In this case, our p-value would be roughly twice the previously calculated p-value.

We could calculate all of those probabilities using our sampling distributions, either simulated or theoretical, that we generated in the previous step. If we chose to calculate a test statistic as defined in the previous section, we could also rely on standard normal distributions to calculate our p-value.

Evaluate your results and write conclusion in context of problem

Once you've gathered your evidence, it's now time to make your final conclusions and determine how you might proceed.

In traditional hypothesis testing, you often make a decision. Recall that you have your threshold (significance level $\alpha$) and your level of evidence (p-value). We can compare the two to determine if your p-value is less than or equal to your threshold. If it is, you have enough evidence to persuade your skeptic to change their mind. If it is larger than the threshold, you don't have quite enough evidence to convince the skeptic.

Common formal conclusions (if given in context) would be:

• I have enough evidence to reject the null hypothesis (the skeptic's claim), and I have sufficient evidence to suggest that the alternative hypothesis is instead true.
• I do not have enough evidence to reject the null hypothesis (the skeptic's claim), and so I do not have sufficient evidence to suggest the alternative hypothesis is true.

The only decision that we can make is to either reject or fail to reject the null hypothesis (we cannot "accept" the null hypothesis). Because we aren't actively evaluating the alternative hypothesis, we don't want to make definitive decisions based on that hypothesis. However, when it comes to making our conclusion for what to use going forward, we frame this on whether we could successfully convince someone of the alternative hypothesis.

A less formal conclusion might look something like:

Based on our sample of Chicago Airbnb listings, it seems as if the mean time since a host has been on Airbnb (for all Chicago Airbnb listings) is more than 5.75 years.

Significance Level Interpretation

We've now seen how the significance level $\alpha$ is used as a threshold for hypothesis testing. What exactly is the significance level?

The significance level $\alpha$ has two primary definitions. One is that the significance level is the maximum probability required to reject the null hypothesis; this is based on how the significance level functions within the hypothesis testing framework. The second definition is that this is the probability of rejecting the null hypothesis when the null hypothesis is true; in other words, this is the probability of making a specific type of error called a Type I error.

Why do we have to be comfortable making a Type I error? There is always a chance that the skeptic was originally correct and we obtained a very unusual sample. We don't want to the skeptic to be so convinced of their theory that no evidence can convince them. In this case, we need the skeptic to be convinced as long as the evidence is strong enough . Typically, the probability threshold will be low, to reduce the number of errors made. This also means that a decent amount of evidence will be needed to convince the skeptic to abandon their position in favor of the alternative theory.

p-value Limitations and Misconceptions

In comparison to the $\alpha$ significance level, we also need to calculate the evidence against the null hypothesis with the p-value.

The p-value is the probability of getting a test statistic as extreme or more extreme (in the direction of the alternative hypothesis), assuming the null hypothesis is true.

Recently, p-values have gotten some bad press in terms of how they are used. However, that doesn't mean that p-values should be abandoned, as they still provide some helpful information. Below, we'll describe what p-values don't mean, and how they should or shouldn't be used to make decisions.

Factors that affect a p-value

What features affect the size of a p-value?

• the null value, or the value assumed under the null hypothesis
• the effect size (the difference between the null value under the null hypothesis and the true value of the parameter)
• the sample size

More evidence against the null hypothesis will be obtained if the effect size is larger and if the sample size is larger.

Misconceptions

We gave a definition for p-values above. What are some examples that p-values don't mean?

• A p-value is not the probability that the null hypothesis is correct
• A p-value is not the probability that the null hypothesis is incorrect
• A p-value is not the probability of getting your specific sample
• A p-value is not the probability that the alternative hypothesis is correct
• A p-value is not the probability that the alternative hypothesis is incorrect
• A p-value does not indicate the size of the effect

Our p-value is a way of measuring the evidence that your sample provides against the null hypothesis, assuming the null hypothesis is in fact correct.

Using the p-value to make a decision

Why is there bad press for a p-value? You may have heard about the standard $\alpha$ level of 0.05. That is, we would be comfortable with rejecting the null hypothesis once in 20 attempts when the null hypothesis is really true. Recall that we reject the null hypothesis when the p-value is less than or equal to the significance level.

Consider what would happen if you have two different p-values: 0.049 and 0.051.

In essence, these two p-values represent two very similar probabilities (4.9% vs. 5.1%) and very similar levels of evidence against the null hypothesis. However, when we make our decision based on our threshold, we would make two different decisions (reject and fail to reject, respectively). Should this decision really be so simplistic? I would argue that the difference shouldn't be so severe when the sample statistics are likely very similar. For this reason, I (and many other experts) strongly recommend using the p-value as a measure of evidence and including it with your conclusion.

Putting too much emphasis on the decision (and having a significant result) has created a culture of misusing p-values. For this reason, understanding your p-value itself is crucial.

Searching for p-values

The other concern with setting a definitive threshold of 0.05 is that some researchers will begin performing multiple tests until finding a p-value that is small enough. However, with a p-value of 0.05, we know that we will have a p-value less than 0.05 1 time out of every 20 times, even when the null hypothesis is true.

This means that if researchers start hunting for p-values that are small (sometimes called p-hacking), then they are likely to identify a small p-value every once in a while by chance alone. Researchers might then publish that result, even though the result is actually not informative. For this reason, it is recommended that researchers write a definitive analysis plan to prevent performing multiple tests in search of a result that occurs by chance alone.

Best Practices

With all of this in mind, what should we do when we have our p-value? How can we prevent or reduce misuse of a p-value?

• Report the p-value along with the conclusion
• Specify the effect size (the value of the statistic)
• Define an analysis plan before looking at the data
• Interpret the p-value clearly to specify what it indicates
• Consider using an alternate statistical approach, the confidence interval, discussed next, when appropriate

Chapter 10: Inference for Means

Hypothesis test for a population mean (1 of 5), learning objectives.

• Recognize when to use a hypothesis test or a confidence interval to draw a conclusion about a population mean.
• Under appropriate conditions, conduct a hypothesis test about a population mean. State a conclusion in context.

Introduction

In Inference for Means , our focus is on inference when the variable is quantitative, so the parameters and statistics are means. In “Estimating a Population Mean,” we learned how to use a sample mean to calculate a confidence interval. The confidence interval estimates a population mean. In “Hypothesis Test for a Population Mean,” we learn to use a sample mean to test a hypothesis about a population mean.

We did hypothesis tests in earlier modules. In Inference for One Proportion , each claim involved a single population proportion. In Inference for Two Proportions , the claim was a statement about a treatment effect or a difference in population proportions. In “Hypothesis Test for a Population Mean,” the claims are statements about a population mean. But we will see that the steps and the logic of the hypothesis test are the same. Before we get into the details, let’s practice identifying research questions and studies that involve a population mean.

Learn By Doing

Cell phone data.

Cell phones and cell phone plans can be very expensive, so consumers must think carefully when choosing a cell phone and service. This decision is as much about choosing the right cellular company as it is about choosing the right phone. Many people use the data/Internet capabilities of a phone as much as, if not more than, they use voice capability. The data service of a cell company is therefore an important factor in this decision. In the following example, a student named Melanie from Los Angeles applies what she learned in her statistics class to help her make a decision about buying a data plan for her smartphone.

Melanie read an advertisement from the Cell Phone Giants (CPG, for short, and yes, we’re using a fictitious company name) that she thinks is too good to be true. The CPG ad states that customers in Los Angeles get average data download speeds of 4 Mbps. With this speed, the ad claims, it takes, on average, only 12 seconds to download a typical 3-minute song from iTunes.

Only 12 seconds on average to download a 3-minute song from iTunes! Melanie has her doubts about this claim, so she gathers data to test it. She asks a friend who uses the CPG plan to download a song, and it takes 13 seconds to download a 3-minute song using the CPG network. Melanie decides to gather more evidence. She uses her friend’s phone and times the download of the same 3-minute song from various locations in Los Angeles. She gets a mean download time of 13.5 seconds for her sample of downloads.

What can Melanie conclude? Her sample has a mean download time that is greater than 12 seconds. Isn’t this evidence that the CPG claim is wrong? Why is a hypothesis test necessary? Isn’t the conclusion clear?

Let’s review the reason Melanie needs to do a hypothesis test before she can reach a conclusion.

Why should Melanie do a hypothesis test?

Melanie’s data (with a mean of 13.5 seconds) suggest that the average download time overall is greater than the 12 seconds claimed by the manufacturer. But wait. We know that samples will vary. If the CPG claim is correct, we don’t expect all samples to have a mean download time exactly equal to 12 seconds. There will be variability in the sample means. But if the overall average download time is 12 seconds, how much variability in sample means do we expect to see? We need to determine if the difference Melanie observed can be explained by chance.

We have to judge Melanie’s data against random samples that come from a population with a mean of 12. For this reason, we must do a simulation or use a mathematical model to examine the sampling distribution of sample means. Based on the sampling distribution, we ask, Is it likely that the samples will have mean download times that are greater than 13.5 seconds if the overall mean is 12 seconds? This probability (the P-value) determines whether Melanie’s data provides convincing evidence against the CPG claim.

Now let’s do the hypothesis test.

Step 1: Determine the hypotheses.

As always, hypotheses come from the research question. The null hypothesis is a hypothesis that the population mean equals a specific value. The alternative hypothesis reflects our claim. The alternative hypothesis says the population mean is “greater than” or “less than” or “not equal to” the value we assume is true in the null hypothesis.

Melanie’s hypotheses:

• H 0 : It takes 12 seconds on average to download Melanie’s song from iTunes with the CPG network in Los Angeles.
• H a : It takes more than 12 seconds on average to download Melanie’s song from iTunes using the CPG network in Los Angeles.

We can write the hypotheses in terms of µ. When we do so, we should always define µ. Here μ = the average number of seconds it takes to download Melanie’s song on the CPG network in Los Angeles.

• H 0 : μ = 12
• H a : μ > 12

Step 2: Collect the data.

To conduct a hypothesis test, Melanie knows she has to use a t-model of the sampling distribution. She thinks ahead to the conditions required, which helps her collect a useful sample.

Recall the conditions for use of a t-model.

• There is no reason to think the download times are normally distributed (they might be, but this isn’t something Melanie could know for sure). So the sample has to be large (more than 30).
• The sample has to be random. Melanie decides to use one phone but randomly selects days, times, and locations in Los Angeles.

Melanie collects a random sample of 45 downloads by using her friend’s phone to download her song from iTunes according to the randomly selected days, times, and locations.

Melanie’s sample of size 45 downloads has an average download time of 13.5 seconds. The standard deviation for the sample is 3.2 seconds. Now Melanie needs to determine how unlikely this data is if CPG’s claim is actually true.

Step 3: Assess the evidence.

Assuming the average download time for Melanie’s song is really 12 seconds, what is the probability that 45 random downloads of this song will have a mean of 13.5 seconds or more?

This is a question about sampling variability. Melanie must determine the standard error. She knows the standard error of random sample means is [latex]\sigma \text{}/\sqrt{n}[/latex]. Since she has no way of knowing the population standard deviation, σ, Melanie uses the sample standard deviation, s = 3.2, as an approximation. Therefore, Melanie approximates the standard error of all sample means ( n = 45) to be

[latex]s\text{}/\sqrt{n}\text{}=\text{}3.2\text{}/\sqrt{45}\text{}=\text{}0.48[/latex]

Now she can assess how far away her sample is from the claimed mean in terms of standard errors. That is, she can compute the t-score of her sample mean.

[latex]T\text{}=\text{}\frac{\mathrm{statistic}-\mathrm{parameter}}{\mathrm{standard}\text{}\mathrm{error}}\text{}=\text{}\frac{\stackrel{¯}{x}-μ}{s\text{}/\sqrt{n}}\text{}=\text{}\frac{13.5-12}{0.48}\text{}=\text{}3.14[/latex]

The sample mean for Melanie’s random sample is approximately 3.14 standard errors above the overall mean of 12. We know from previous experience that a sample mean this far above µ is very unlikely. With a t-score this large, the P-value is very small. We use a simulation of the t-model for 44 degrees of freedom to verify this.

We want the probability that the sample mean is greater than 13.5. This corresponds to the probability that T is greater than 3.14. The P-value is 0.0015.

Step 4: State a conclusion.

Here the logic is the same as for other hypothesis tests. We use the P-value to make a decision. The P-value helps us determine if the difference we see between the data and the hypothesized value of µ is statistically significant or due to chance. One of two outcomes can occur:

• One possibility is that results similar to the actual sample are extremely unlikely. This means the data does not fit with results from random samples selected from the population described by the null hypothesis. In this case, it is unlikely that the data came from this population. The probability as measured by the P-value is small, so we view this as strong evidence against the null hypothesis. We reject the null hypothesis in favor of the alternative hypothesis.
• The other possibility is that results similar to the actual sample are fairly likely (not unusual). This means the data fits with typical results from random samples selected from the population described by the null hypothesis. The probability as measured by the P-value is large. In this case, we do not have evidence against the null hypothesis, so we cannot reject it in favor of the alternative hypothesis.

Melanie’s data is very unlikely if µ = 12. The probability is essentially zero (P-value = 0.0015). This means we will rarely see sample means greater than 13.5 if µ = 12. So we reject the null and accept the alternative hypothesis. In other words, this sample provides strong evidence that CPG has overstated the speed of its data download capability.

The following activities give you an opportunity to practice parts of the hypothesis testing process for a population mean. Later you will have the opportunity to practice the hypothesis test from start to finish.

For the following scenarios, give the null and alternative hypotheses and state in words what µ represents in your hypotheses. A good definition of µ describes both the variable and the population.

In the previous example, Melanie did not state a significance level for her test. If she had, the logic is the same as we used for hypothesis tests in Modules 8 and 9. To come to a conclusion about H 0 , we compare the P-value to the significance level α.

• If P ≤ α, we reject H 0 . We conclude there is significant evidence in favor of H a .
• If P > α, we fail to reject H 0 . We conclude the sample does not provide significant evidence in favor of H a .

Use this simulation when needed to answer questions below.

• Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution

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5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

• At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)). 
• The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
• The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

Statistics Made Easy

4 Examples of Hypothesis Testing in Real Life

In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

• Null Hypothesis (H 0 ): The sample data occurs purely from chance.
• Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that we have sufficient evidence to say that the alternative hypothesis is true.

The following examples provide several situations where hypothesis tests are used in the real world.

Example 1: Biology

Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals.

For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test using the following hypotheses:

• H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
• H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

If the p-value of the test is less than some significance level (e.g. α = .05), then she can reject the null hypothesis and conclude that the fertilizer leads to increased plant growth.

Example 2: Clinical Trials

Hypothesis tests are often used in clinical trials to determine whether some new treatment, drug, procedure, etc. causes improved outcomes in patients.

For example, suppose a doctor believes that a new drug is able to reduce blood pressure in obese patients. To test this, he may measure the blood pressure of 40 patients before and after using the new drug for one month.

He then performs a hypothesis test using the following hypotheses:

• H 0 : μ after = μ before (the mean blood pressure is the same before and after using the drug)
• H A : μ after < μ before (the mean blood pressure is less after using the drug)

If the p-value of the test is less than some significance level (e.g. α = .05), then he can reject the null hypothesis and conclude that the new drug leads to reduced blood pressure.

Example 3: Advertising Spend

Hypothesis tests are often used in business to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales.

For example, suppose a company believes that spending more money on digital advertising leads to increased sales. To test this, the company may increase money spent on digital advertising during a two-month period and collect data to see if overall sales have increased.

They may perform a hypothesis test using the following hypotheses:

• H 0 : μ after = μ before (the mean sales is the same before and after spending more on advertising)
• H A : μ after > μ before (the mean sales increased after spending more on advertising)

If the p-value of the test is less than some significance level (e.g. α = .05), then the company can reject the null hypothesis and conclude that increased digital advertising leads to increased sales.

Example 4: Manufacturing

Hypothesis tests are also used often in manufacturing plants to determine if some new process, technique, method, etc. causes a change in the number of defective products produced.

For example, suppose a certain manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, they may measure the mean number of defective widgets produced before and after using the new method for one month.

They can then perform a hypothesis test using the following hypotheses:

• H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
• H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

If the p-value of the test is less than some significance level (e.g. α = .05), then the plant can reject the null hypothesis and conclude that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

Introduction to Hypothesis Testing Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test

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How to Write a Great Hypothesis

Hypothesis Definition, Format, Examples, and Tips

Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk,  "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.

Verywell / Alex Dos Diaz

• The Scientific Method

Hypothesis Format

Falsifiability of a hypothesis.

• Operationalization

Hypothesis Types

Hypotheses examples.

• Collecting Data

Frequently Asked Questions

A  hypothesis  is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.

Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."

At a Glance

A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.

The Hypothesis in the Scientific Method

In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:

• Forming a question
• Performing background research
• Creating a hypothesis
• Designing an experiment
• Collecting data
• Analyzing the results
• Drawing conclusions
• Communicating the results

The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.

Unless you are creating an exploratory study, your hypothesis should always explain what you  expect  to happen.

In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.

Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.

In many cases, researchers may find that the results of an experiment  do not  support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.

In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."

In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."

Elements of a Good Hypothesis

So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:

• Is your hypothesis based on your research on a topic?
• Can your hypothesis be tested?
• Does your hypothesis include independent and dependent variables?

Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the  journal articles you read . Many authors will suggest questions that still need to be explored.

How to Formulate a Good Hypothesis

To form a hypothesis, you should take these steps:

• Collect as many observations about a topic or problem as you can.
• Evaluate these observations and look for possible causes of the problem.
• Create a list of possible explanations that you might want to explore.
• After you have developed some possible hypotheses, think of ways that you could confirm or disprove each hypothesis through experimentation. This is known as falsifiability.

In the scientific method ,  falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.

Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that  if  something was false, then it is possible to demonstrate that it is false.

One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.

The Importance of Operational Definitions

A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.

Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.

For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.

These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.

Replicability

One of the basic principles of any type of scientific research is that the results must be replicable.

Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.

Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.

To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.

Hypothesis Checklist

• Does your hypothesis focus on something that you can actually test?
• Does your hypothesis include both an independent and dependent variable?
• Can you manipulate the variables?
• Can your hypothesis be tested without violating ethical standards?

The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:

• Simple hypothesis : This type of hypothesis suggests there is a relationship between one independent variable and one dependent variable.
• Complex hypothesis : This type suggests a relationship between three or more variables, such as two independent and dependent variables.
• Null hypothesis : This hypothesis suggests no relationship exists between two or more variables.
• Alternative hypothesis : This hypothesis states the opposite of the null hypothesis.
• Statistical hypothesis : This hypothesis uses statistical analysis to evaluate a representative population sample and then generalizes the findings to the larger group.
• Logical hypothesis : This hypothesis assumes a relationship between variables without collecting data or evidence.

A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the  dependent variable  if you change the  independent variable .

The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."

A few examples of simple hypotheses:

• "Students who eat breakfast will perform better on a math exam than students who do not eat breakfast."
• "Students who experience test anxiety before an English exam will get lower scores than students who do not experience test anxiety."​
• "Motorists who talk on the phone while driving will be more likely to make errors on a driving course than those who do not talk on the phone."
• "Children who receive a new reading intervention will have higher reading scores than students who do not receive the intervention."

Examples of a complex hypothesis include:

• "People with high-sugar diets and sedentary activity levels are more likely to develop depression."
• "Younger people who are regularly exposed to green, outdoor areas have better subjective well-being than older adults who have limited exposure to green spaces."

Examples of a null hypothesis include:

• "There is no difference in anxiety levels between people who take St. John's wort supplements and those who do not."
• "There is no difference in scores on a memory recall task between children and adults."
• "There is no difference in aggression levels between children who play first-person shooter games and those who do not."

Examples of an alternative hypothesis:

• "People who take St. John's wort supplements will have less anxiety than those who do not."
• "Adults will perform better on a memory task than children."
• "Children who play first-person shooter games will show higher levels of aggression than children who do not." 

Collecting Data on Your Hypothesis

Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.

Descriptive Research Methods

Descriptive research such as  case studies ,  naturalistic observations , and surveys are often used when  conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.

Once a researcher has collected data using descriptive methods, a  correlational study  can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.

Experimental Research Methods

Experimental methods  are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).

Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually  cause  another to change.

The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.

Some examples of how to write a hypothesis include:

• "Staying up late will lead to worse test performance the next day."
• "People who consume one apple each day will visit the doctor fewer times each year."
• "Breaking study sessions up into three 20-minute sessions will lead to better test results than a single 60-minute study session."

The four parts of a hypothesis are:

(1) The research question

(2) The independent variable (IV)

(3) The dependent variable (DV)

(4) The proposed relationship between the IV and DV

No, a hypothesis and a theory are not the same thing. A hypothesis is a testable prediction about a specific research question. A theory, on the other hand, is an explanation supported by an existing body of scientific research.

Thompson WH, Skau S. On the scope of scientific hypotheses .  R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607

Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:].  Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z

Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004

Nosek BA, Errington TM. What is replication ?  PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691

Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies .  Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18

Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.

By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."

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• Knowledge Base

Methodology

• Population vs. Sample | Definitions, Differences & Examples

Population vs. Sample | Definitions, Differences & Examples

Published on May 14, 2020 by Pritha Bhandari . Revised on June 21, 2023.

A population is the entire group that you want to draw conclusions about.

A sample is the specific group that you will collect data from. The size of the sample is always less than the total size of the population.

In research, a population doesn’t always refer to people. It can mean a group containing elements of anything you want to study, such as objects, events, organizations, countries, species, organisms, etc.

Table of contents

Collecting data from a population, collecting data from a sample, population parameter vs. sample statistic, practice questions : populations vs. samples, other interesting articles, frequently asked questions about samples and populations.

Populations are used when your research question requires, or when you have access to, data from every member of the population.

Usually, it is only straightforward to collect data from a whole population when it is small, accessible and cooperative.

For larger and more dispersed populations, it is often difficult or impossible to collect data from every individual. For example, every 10 years, the federal US government aims to count every person living in the country using the US Census. This data is used to distribute funding across the nation.

However, historically, marginalized and low-income groups have been difficult to contact, locate and encourage participation from. Because of non-responses, the population count is incomplete and biased towards some groups, which results in disproportionate funding across the country.

In cases like this, sampling can be used to make more precise inferences about the population.

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When your population is large in size, geographically dispersed, or difficult to contact, it’s necessary to use a sample. With statistical analysis , you can use sample data to make estimates or test hypotheses about population data.

Ideally, a sample should be randomly selected and representative of the population. Using probability sampling methods (such as simple random sampling or stratified sampling ) reduces the risk of sampling bias and enhances both internal and external validity .

For practical reasons, researchers often use non-probability sampling methods. Non-probability samples are chosen for specific criteria; they may be more convenient or cheaper to access. Because of non-random selection methods, any statistical inferences about the broader population will be weaker than with a probability sample.

Reasons for sampling

• Necessity : Sometimes it’s simply not possible to study the whole population due to its size or inaccessibility.
• Practicality : It’s easier and more efficient to collect data from a sample.
• Cost-effectiveness : There are fewer participant, laboratory, equipment, and researcher costs involved.
• Manageability : Storing and running statistical analyses on smaller datasets is easier and reliable.

When you collect data from a population or a sample, there are various measurements and numbers you can calculate from the data. A parameter is a measure that describes the whole population. A statistic is a measure that describes the sample.

You can use estimation or hypothesis testing to estimate how likely it is that a sample statistic differs from the population parameter.

Sampling error

A sampling error is the difference between a population parameter and a sample statistic. In your study, the sampling error is the difference between the mean political attitude rating of your sample and the true mean political attitude rating of all undergraduate students in the Netherlands.

Sampling errors happen even when you use a randomly selected sample. This is because random samples are not identical to the population in terms of numerical measures like means and standard deviations .

Because the aim of scientific research is to generalize findings from the sample to the population, you want the sampling error to be low. You can reduce sampling error by increasing the sample size.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Student’s  t -distribution
• Normal distribution
• Null and Alternative Hypotheses
• Chi square tests
• Confidence interval
• Cluster sampling
• Stratified sampling
• Data cleansing
• Reproducibility vs Replicability
• Peer review
• Likert scale

Research bias

• Implicit bias
• Framing effect
• Cognitive bias
• Placebo effect
• Hawthorne effect
• Hindsight bias
• Affect heuristic

Samples are used to make inferences about populations . Samples are easier to collect data from because they are practical, cost-effective, convenient, and manageable.

Populations are used when a research question requires data from every member of the population. This is usually only feasible when the population is small and easily accessible.

A statistic refers to measures about the sample , while a parameter refers to measures about the population .

A sampling error is the difference between a population parameter and a sample statistic .

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Bhandari, P. (2023, June 21). Population vs. Sample | Definitions, Differences & Examples. Scribbr. Retrieved April 15, 2024, from https://www.scribbr.com/methodology/population-vs-sample/

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