How to Do 1-Sample t-Tests: Steps & Tutorial

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

In this lesson, you will learn how to calculate the ''t''-score for a set of data and use it to determine whether the sample mean and population mean are statistically different.

Steps of a t-Test

T-tests can be used to compare the mean of a sample to the mean of an entire population.

Step 1: Determine the sample mean, population mean, sample standard deviation and sample size for the data. Calculate any values that are not provided.

Step 2: Calculate the t-score for the data using the t-score formula.


T-score formula
tscore

Step 3: Identify the critical t-score. If it is not provided, it can be found on a t-score table, based on the degree of freedom and alpha value. The degree of freedom is one less than the sample size. Unless specified otherwise, 0.05 should be used for the alpha value.

Step 4: Compare the calculated t-score to the critical t-score. If the calculated t-score is greater than the critical t-score, then the sample mean is statistically different than the population mean. Otherwise, there is no statistical difference between the sample and population means.

Example: Comparing Test Scores

The following data represents the math ACT scores for a sample of twenty students at a college. The mean ACT score for students enrolled is 26. Using a critical value of 2.086, determine if the score of the sample of students is different from the college population's score.

21 21 23 23
24 25 26 26
26 27 27 28
28 29 30 31
31 31 33 34
  • Step 1: Determine the sample mean, population mean, sample standard deviation and sample size.

The sample size is the number of scores in the sample, which is twenty. The population mean is 26, which was given in the problem. The sample mean can be calculated by adding all of the scores in the sample and dividing by the sample size.

(21 + 21 + 23 + 23 + 24 + 25 + 26 + 26 + 26 + 27 + 27 + 28 + 28 + 29 + 30 + 31 + 31 + 31 + 33 + 34) / 20 = 27.2

The sample standard deviation can be found using the following formula:


Sample standard deviation formula
ssd

In this formula, x represents each value in the sample, which would be subtracted by the mean and squared. Since this must be done for each value, it is useful to organize it in a table.

(21 - 27.2)2 = 38.44 (21 - 27.2)2 = 38.44 (23 - 27.2)2 = 17.64 (23 - 27.2)2 = 17.64
(24 - 27.2)2 = 10.24 (25 - 27.2)2 = 4.84 (26 - 27.2)2 = 1.44 (26 - 27.2)2 = 1.44
(26 - 27.2)2 = 1.44 (27 - 27.2)2 = 0.04 (27 - 27.2)2 = 0.04 (28 - 27.2)2 = 0.64
(28 - 27.2)2 = 0.64 (29 - 27.2)2 = 3.24 (30 - 27.2)2 = 7.84 (31 - 27.2)2 = 14.44
(31 - 27.2)2 = 14.44 (31 - 27.2)2 = 14.44 (33 - 27.2)2 = 33.64 (34 - 27.2)2 = 46.24

The sum of these values is 267.2. The variable n in the formula represents the sample size, which is twenty. Substituting these into the formula gives us 3.75 for the sample standard deviation.


Sample standard deviation calculation
ssd1

  • Step 2: Calculate the t-score for the data using the t-score formula.

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