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

The following examples show how to perform the three types of one sample t-tests:

- Two-tailed one sample t-test
- Right-tailed one sample t-test
- Left-tailed one sample t-test

Let’s jump in!

**Example 1: Two-Tailed One Sample T-Test**

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.

**Example 2: Right-Tailed One Sample T-Test**

Suppose we suspect that the mean exam score on a certain college entrance exam is greater than the accepted mean score of 82.

To test this, will perform a right-tailed 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 exam scores with the following information:

- Sample size n = 60
- Sample mean x = 84
- Sample standard deviation s = 8.1

**Step 2: Define the hypotheses.**

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

**H**μ ≤ 82_{0}:**H**μ > 82_{1}:

**Step 3: Calculate the test statistic t.**

**t **= (x – μ) / (s/√n) = (84-82) / (8.1/√60) = **1.9125**

**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 = 1.9125 and degrees of freedom = n-1 = 60-1 = 59 is **0.0303**.

**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 exam score on this particular exam is greater than 82.

**Example 3: Left-Tailed One Sample T-Test**

Suppose we suspect that the mean height of a particular species of plant is less than the accepted mean height of 10 inches.

To test this, will perform a left-tailed 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 plants with the following information:

- Sample size n = 25
- Sample mean x = 9.5
- Sample standard deviation s = 3.5

**Step 2: Define the hypotheses.**

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

**H**μ ≥ 10_{0}:**H**μ < 10_{1}:

**Step 3: Calculate the test statistic t.**

**t **= (x – μ) / (s/√n) = (9.5-10) / (3.5/√25) = **-0.7143**

**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 = -0.7143 and degrees of freedom = n-1 = 25-1 = 24 is **0.24097**.

**Step 5: Draw a conclusion.**

Since this p-value is not less than our significance level α = 0.05, we fail to reject the null hypothesis. We do not have sufficient evidence to say that the mean height for this particular plant species is less than 10 inches.

**Additional Resources**

The following tutorials provide additional information about hypothesis testing:

Introduction to the One Sample t-test

One Sample t-test Calculator

How to Conduct a One Sample t-test in Excel