Final answer:
To perform the hypothesis test at the 1% significance level with the given sample data and null hypothesis, a one-mean t-test using the Student's t-distribution is appropriate. You calculate the test statistic with the given formula and compare it against the critical t value. Depending on the result, you either reject or fail to reject the null hypothesis.
Step-by-step explanation:
To perform a hypothesis test at the 1% significance level for the given sample data (sample mean: x = 22, sample standard deviation: s = 6, sample size: n = 24), and the null hypothesis (H0: μ = 21), you would use a one-mean t-test. Since we do not have the population standard deviation, the Student's t-distribution is appropriate.
First, we check if the underlying population is assumed to be normal, which is a requirement for the t-test. The sample size also needs to be considered, and since the sample size is less than 30, and there is no indication that the population is not normal, the t-distribution is suitable. To calculate the test statistic (t), we use the formula:
t = (x - μ) / (s/ √ n)
Using our data, we would calculate this as t = (22 - 21) / (6 / √ 24), which would give us the value of t. We then compare the calculated t value to the critical t value from the t-distribution table at n-1 degrees of freedom (in this case, 23) and a 1% significance level for a two-tailed test (since no direction is specified).
If the absolute value of the calculated t is greater than the critical t value, we reject the null hypothesis, otherwise, we fail to reject it.