Question

Using R, carry out the appropriate hypothesis test, and provide

(Round your final answer to four decimal places. Assume unequal variances.)

a) the P-value of your test.

b) the numerical value of the test statistic.

Answer 1

The student is asked to perform a hypothesis test comparing two variances, specifically using an F-test. The null and alternative hypotheses involve equality versus inequality of variances of the two students' test scores. Using sample standard deviations and calculating an F statistic and p-value with a significance level of 0.05 determines whether to reject the null hypothesis.

Step-by-step explanation:

In the context of the provided information, the student is engaged in conducting a hypothesis test comparing two variances or standard deviations, which means they should use an F-test. Here's how to address the different components:

1. State an appropriate null hypothesis: This involves specifying that there is no difference in the variance of the two populations. For the given scenario, the null hypothesis (H0) could be: σ²1 = σ²2, indicating that the variance of the first student's test scores is equal to the variance of the second student's test scores.
2. State an appropriate alternative hypothesis: The alternative hypothesis (Ha) would indicate that there is a difference in variance, which could be expressed as σ²1 ≠ σ²2.
3. Define the test statistic: The F statistic is used for comparing the ratios of variances and is calculated based on the sample standard deviations provided.
4. Calculate the test statistic: Use the formula F = s²1/s²2, where s²1 and s²2 are the sample variances. In this case, F = (38.1)2 / (22.5)2.
5. Calculate the p-value: This can be found using an F-distribution table or software like R, comparing the calculated F statistic with the theoretical F-distribution.
6. Decision about the null hypothesis: If the p-value is less than the significance level (α = 0.05), the null hypothesis is rejected, indicating that there is evidence to suggest the variances are different.
7. Errors: A Type I error would occur if the null hypothesis is wrongly rejected (false positive). A Type II error would occur if the null hypothesis is wrongly not rejected (false negative).

In R, the F-test can be performed using the tartest() function to compare two variances. Following the calculation of the F statistic and p-value, one can decide to reject the null hypothesis if the p-value is less than 0.05, thereby suggesting that the second student's claim of having more consistent scores may be statistically justified.