In statistical tests, statistical significance is determined by citing an alpha level, or the probability of rejecting the null hypothesis when the null hypothesis is true. For this example, alpha, or significance level, is set to 0.05 (5%).
The formula for the t-test is as follows.
In this equation, x̄ is the sample mean, μ is the population mean, s is the sample standard deviation, and n is the number of observations in the sample. The sample mean (75), the sample standard deviation (9.3), and the number of observations in the sample (9) are all known. Assume the average height of students in the school is 69 inches:
The calculated t-value can be used to test the original hypotheses and determine statistical significance. The first step is to look at a t-table and find the value associated with 8 degrees of freedom (sample size – 1) and our alpha level of 0.05. Because the test determines statistical difference between sample mean (class) and population mean (class), this is considered a two-tailed test. For this reason, the alpha level is divided in half (0.05/2 = 0.025) and then located on the t-table to find our critical value, which comes out to be 2.306. Because the t-value is lower than the critical value on the t-table, we fail to reject the null hypothesis that the sample mean and population mean are statistically different at the 0.05 significance level.
The other statistical test that could be used is a z-test, but this test is only appropriate when the sample size is above 30 and the standard deviation of the population is known. The formula for the z-test is the same as the t-test formula.