The student academic group on a college campus claims that second-year students study around 2.7 hours per day, on average. However, one Introduction to Statistics class at Emory believed that the average would be significantly greater than 2.7 hours per day, so they took a random sample of 28 second-year students and performed a two-sided hypothesis test. At 𝛼 = 0.01, which of the following represents the smallest test statistic required to reject the null hypothesis? You must use the t-table to answer this question. Also, treat the test statistic as is; Do NOT round up or down the numbers below.

Question

BlackTom AI · Accepted Answer

To determine the smallest test statistic that would lead to rejecting the null hypothesis at alpha = 0.01 in a two-sided t-test, we first identify the critical t-value from the t-distribution with the appropriate degrees of freedom. Here, n = 28, so df = n − 1 = 27. For a two-sided test with alpha = 0.01, each tail has 0.005, so we look up t_{0.005, 27} on the t-table. The critical boundary is approximately 2.77, specifically around 2.771, which is the magnitude required to reject H0.

Option by option analysis:
- 2.475: This value is well below the critical 2.771; since the test statistic does not reach the threshold, the null hypothesis would not be rejected at the 1% level.
- 2.765: This is very close to the critical value but still slightly below 2.771. As such, it does not surpass the cutoff, so we would fail to reject H0 at alpha = 0.01.
- 2.775: This number is just above the critical boundary of 2.771, so it crosses the threshold needed to reject H0. In other words, any test statistic with magnitude at least this large would lead to rejection at the 1% level for df = 27.
- 2.875: Although this value also exceeds the critical value and would lead to rejection, it is not the smallest statistic among the options that would do so.

Thus, considering the requirement to use the t-table value for df = 27 at a two-sided 0.01 level and comparing each option to the cutoff, the smallest statistic that would result in rejection is the one that just surpasses the critical t-value.

类似问题

The critical value t 𝛼 2 ,k is the (1- 𝛼 /2) - quantile for a t distribution with k degrees of freedom. Compute t 𝛼 2 ,k when 𝛼 = .02 and k = 17

Which of the following is true regarding the t-distribution?

This time, let's construct a 95% confidence interval for a true mean price difference between a 0.99 carat and a 1 carat diamond. What is the correct t-score for the margin of error?

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