Question

The state of Oregon would like to test whether the average GPA of college students is different than 3.2 or not for scholarship purposes. They take a random sample of 1000 college students in Oregon. Here is the summary statistics.

Sample mean, x ¯	3.18
Sample size, n	10,000
Sample standard deviation, s	0.46

Refer to the information in the previous problem.

Suppose we test like to test the hypotheses,

H₀ : μ = 3.2  vs.  H_a: μ ≠ 3.2

with a significance level of α = 0.05. Which of the following methods are a valid way to test the hypotheses?

For each of the following methods indicate whether they are a valid way to test the hypotheses. In other words, is the method described an appropriate way of finding a p-value and trusting the conclusion is a correct conclusion considering the data.

Method 1. Using our experience on the subject, we try to guess what the p-value is likely to be.  

Method 1 is                            [ Select ]                       ["a valid", "NOT a valid"]           procedure.

Method 2. Take 10000 samples from the population of all Oregon college students, for each sample record the mean. Find the p-value by determining the proportion of these means that are greater than different than 3.2.

Method 2 is                            [ Select ]                       ["a valid", "NOT a valid"]           procedure.

Method 3. Calculate the 95% confidence interval for μ , if the null hypothesized value does not fall within the interval then we can reject the null hypothesis.

Method 3 is                            [ Select ]                       ["a valid", "NOT a valid"]           procedure.

Method 4. Perform a one sample z-test for the proportion of GPAs greater than 3.2

Method 4 is                            [ Select ]                       ["a valid", "NOT a valid"]           procedure.

Method 5. To solve for the p-value, they can calculate a t-test statistic and degrees of freedom and find the area under the t-distribution curve that is more unusual than the calculated test statistic.

Method 5 is                            [ Select ]                       ["a valid", "NOT a valid"]           procedure.

Recall that the alternative hypothesis is H_a: μ ≠ 3.2 and the sample mean was found to be 3.18 from 10,000 college students from across Oregon. The p-value was determined to be less than 0.0001. With this in mind, which of the following statements is correct?

Group of answer choices

We have found statistical significance but no practical significance.

We have found both statistical and practical significance.

We have found no statistical significance but we do see practical significance in the data.

This study does not have any statistical or practical significance.

Answer 1

Method 1. Using our experience on the subject, we try to guess what the p-value is likely to be.  

Method 1 is "NOT a valid" procedure.

Guessing a p-value is not a valid procedure.

Method 2. Take 10000 samples from the population of all Oregon college students, for each sample record the mean. Find the p-value by determining the proportion of these means that are greater than different than 3.2.

Method 2 is                            "NOT a valid" procedure.

p-value is not the proportion of these means that are greater than different than 3.2.

Method 3. Calculate the 95% confidence interval for μ , if the null hypothesized value does not fall within the interval then we can reject the null hypothesis.

Method 3 is "a valid" procedure.

If the null hypothesized value does not fall within the interval, we are 95% confident (or at 0.05 significance level) that the true mean is not equal to null hypothesized value.

Method 4. Perform a one sample z-test for the proportion of GPAs greater than 3.2

Method 4 is                                               "NOT a valid" procedure.

We need to perform test for the mean of GPAs not equal to 3.2

Method 5. To solve for the p-value, they can calculate a t-test statistic and degrees of freedom and find the area under the t-distribution curve that is more unusual than the calculated test statistic.

Method 5 is                      "a valid" procedure.

Since we do not know the population standard deviation, we need to perform one sample t-test. Although sample size is very large (10,000) so applying z and t distributions are equivalent for such large sample size.

Recall that the alternative hypothesis is H_a: μ ≠ 3.2 and the sample mean was found to be 3.18 from 10,000 college students from across Oregon. The p-value was determined to be less than 0.0001. With this in mind, which of the following statements is correct?

Since p-value was less than 0.0001, we reject the null hypothesis and the results are significant. But the difference of 0.02 in GPA's is of no practical significance.

We have found statistical significance but no practical significance.