Final answer:
The t statistic for a one-sample t-test using Student's t-distribution, given the sample mean of 30, sample standard deviation of 5, sample size of 29, and significance level of 0.01, is 5.385. This t statistic is used to test the hypothesis that the population mean is greater than 25, and should be compared to the critical value at the 0.01 significance level with 28 degrees of freedom.
Step-by-step explanation:
The question relates to computing the t statistic for a one-sample t-test to determine whether there is sufficient evidence to conclude that a population mean is greater than a certain value. Given a sample mean (×) of 30, a sample standard deviation (s) of 5, a sample size (n) of 29, and a significance level (α) of 0.01, the hypothesis being tested is whether the population mean is greater than 25. Since the population standard deviation is unknown and the sample size is small (n < 30), we use the Student's t-distribution to calculate the t statistic.
The t statistic is calculated using the formula: t = (× - μ) / (s / √n), where × is the sample mean, μ is the hypothesized population mean, s is the sample standard deviation, and n is the sample size. Plugging in the numbers, we get: t = (30 - 25) / (5 / √29) = 5 / (5 / 5.385) = 5.385. The degrees of freedom for this test are n - 1, which is 29 - 1 = 28. The next step would involve comparing the calculated t statistic to the critical t value from the t distribution table at the 0.01 significance level and 28 degrees of freedom to decide whether to reject the null hypothesis.
Remember that when conducting a hypothesis test, one-tailed or two-tailed tests should be specified based on the alternative hypothesis, which in this case is a one-tailed test since the hypothesis is that the population mean is greater than 25. The calculated t statistic provides the basis to determine if we can reject the null hypothesis of the population mean being equal to or less than 25 at the given level of significance.