A **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:

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:

**H**μ = μ_{0}:_{0}(population mean is equal to some hypothesized value μ_{0})

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

**H**μ ≠ μ_{1}(two-tailed):_{0}(population mean is not equal to some hypothesized value μ_{0})**H**μ < μ_{1}(left-tailed):_{0}(population mean is less than some hypothesized value μ_{0})**H**μ > μ_{1}(right-tailed):_{0}(population mean is greater than some hypothesized value μ_{0})

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

**t = (****x – μ) / (s/√n)**

where:

**x:**sample mean**μ**hypothesized population mean_{0}:**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:

**H**μ = 310 (population mean is equal to 310 pounds)_{0}:**H**μ ≠ 310 (population mean is not equal to 310 pounds)_{1}:

**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