One Sample t-test: Definition, Formula, and Example


one sample t-test is used to test whether or not the mean of a population is equal to some value.

This tutorial explains the following:

  • The motivation for performing a one sample t-test.
  • The formula to perform a one sample t-test.
  • The assumptions that should be met to perform a one sample t-test.
  • An example of how to perform a one sample t-test.

One Sample t-test: Motivation

Suppose we want to know whether or not the mean weight of a certain species of turtle in Florida is equal to 310 pounds. Since there are thousands of turtles in Florida, it would be extremely time-consuming and costly to go around and weigh each individual turtle.

Instead, we might take a simple random sample of 40 turtles and use the mean weight of the turtles in this sample to estimate the true population mean:

Sample from population example

However, it’s virtually guaranteed that the mean weight of turtles in our sample will differ from 310 pounds. The question is whether or not this difference is statistically significant. Fortunately, a one sample t-test allows us to answer this question.

One Sample t-test: Formula

A one-sample t-test always uses the following null hypothesis:

  • H0μ = μ0 (population mean is equal to some hypothesized value μ0)

The alternative hypothesis can be either two-tailed, left-tailed, or right-tailed:

  • H1 (two-tailed): μ ≠ μ0 (population mean is not equal to some hypothesized value μ0)
  • H1 (left-tailed): μ < μ0 (population mean is less than some hypothesized value μ0)
  • H1 (right-tailed): μ > μ0 (population mean is greater than some hypothesized value μ0)

We use the following formula to calculate the test statistic t:

t = (x – μ) / (s/√n)

where:

  • x: sample mean
  • μ0: hypothesized population mean
  • s: sample standard deviation
  • n: sample size

If the p-value that corresponds to the test statistic t with (n-1) degrees of freedom is less than your chosen significance level (common choices are 0.10, 0.05, and 0.01) then you can reject the null hypothesis.

One Sample t-test: Assumptions

For the results of a one sample t-test to be valid, the following assumptions should be met:

  • The variable under study should be either an interval or ratio variable.
  • The observations in the sample should be independent.
  • The variable under study should be approximately normally distributed. You can check this assumption by creating a histogram and visually checking if the distribution has roughly a “bell shape.”
  • The variable under study should have no outliers. You can check this assumption by creating a boxplot and visually checking for outliers.

One Sample t-test: Example

Suppose we want to know whether or not the mean weight of a certain species of turtle is equal to 310 pounds. To test this, will perform a one-sample t-test at significance level α = 0.05 using the following steps:

Step 1: Gather the sample data.

Suppose we collect a random sample of turtles with the following information:

  • Sample size n = 40
  • Sample mean weight x = 300
  • Sample standard deviation s = 18.5

Step 2: Define the hypotheses.

We will perform the one sample t-test with the following hypotheses:

  • H0μ = 310 (population mean is equal to 310 pounds)
  • H1μ ≠ 310 (population mean is not equal to 310 pounds)

Step 3: Calculate the test statistic t.

t = (x – μ) / (s/√n) = (300-310) / (18.5/√40) = -3.4187

Step 4: Calculate the p-value of the test statistic t.

According to the T Score to P Value Calculator, the p-value associated with t = -3.4817 and degrees of freedom = n-1 = 40-1 = 39 is 0.00149.

Step 5: Draw a conclusion.

Since this p-value is less than our significance level α = 0.05, we reject the null hypothesis. We have sufficient evidence to say that the mean weight of this species of turtle is not equal to 310 pounds.

Note: You can also perform this entire one sample t-test by simply using the One Sample t-test calculator.

Additional Resources

The following tutorials explain how to perform a one-sample t-test using different statistical programs:

How to Perform a One Sample t-Test in Excel
How to Perform a One Sample t-test in SPSS
How to Perform a One Sample t-test in Stata
How to Perform a One Sample t-test on a TI-84 Calculator

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