A school board is interested in determining whether elementary students' level of numerical skills is different between in-person and online schools. The school board randomly selects 20 grade 5 students who participated in school in person and 20 grade 5 students who participated in virtual school. All students take a test in mathematics. These are the descriptive statistics of the results:

In-Person Virtual

Mean Score 85 83

Standard Deviation 20 15

Based on this information, answer the following question. Suppose normality assumptions are warranted.

Question 1

We would like to test if in-person and virtual schools are different in the level of numerical skills of their elementary students. Which one is the appropriate hypothesis test?

a) Chi-square test of homogeneity.

b) Two-sample paired t-test.

c) Two-sample t-test.

d) Two-sample z-test.

Question 2

If_Irepresents the mean math score of grade 5 in-person students and _Vrepresents the mean math score of grade 5 virtual students, what are the appropriate hypotheses that the school board should consider?

a) H₀: _I= _Vvs. H_a: _{I V}

b) H₀: _I= _Vvs. H_a: _{I V}

c) H₀: _{I V}vs. H_a: _I= _V

d) H₀: _I> _Vvs. H_a: _{I V}

Question 3

Suppose students in both in-person and virtual schools have a similar background. The school board suggests that the variance of the math scores is the same in the two populations. Calculate the pooled standard deviation.

a) 312.6

b) 15.25

c) 17.67

d) 232.56

Question 4

Regardless of your answer to the previous question, supposed the pooled standard deviation is 18. Assuming equal variances, hat is the standard error of the difference between the mean math score of in-person students and the mean math score of virtual students?

a) 4.42

b) 3.38

c) 5.69

d) 1.34

Question 5

Assuming that the standard error of the difference between the mean math scores is 6, the value of the appropriate test statistic is:

a) t₃₈=0.33

b) t₃₉=0.33

c) t₁₉=0.33

d) z=0.33

Question 6

Assuming that the standard error of the difference between the mean math scores is 6, the 95% confidence interval correspondingto the appropriate test for this study is:

a) (-10.55, 14,55)

b) (-10.68, 14.65)

c) (-8.11, 12.11)

d) (-10.14, 14.14)

Question 7

Assume that the true 95% confidence interval for the difference in mean scores is (-9.31, 13.31). What statement is true?

a) We reject the null hypothesis as the confidence interval captures 2 (the difference between 83 and 85).

b) We fail to reject the null hypothesis as the confidence interval captures zero.

c) We reject the null hypothesis as the confidence interval captures zero.

d) We fail to reject the null hypothesis as the confidence interval captures 2 (the differencebetween 83 and 85).

Question 8

The school board realizes that students attending virtual schools are more self-motivated and thus concludes that the equal variance assumption is not warranted. What would it change in the analysis?

a) A new math test should be designed as the original test assumed students have a similar background.

b) We cannot be sure self-motivation affects the equal variance assumption. Therefore, we don't care what the school board thinks about the previous analysis.

c) We need to see the math scores of 40 students before commenting on what method is suitable for hypothesis testing.

d) A new pooled standard error should be calculated and the degrees of freedom stay at 38.

e) A new standard error should be calculated and the degree of freedom should be calculated as well.

Question 9

In light of new information, now suppose the histogram of math scores of in-person students and virtual classes are Bimodal. What would you choose for hypothesis testing?

a) We pay attention to the bimodal histogram of math scores and use a non-parametric test, even though non-parametric tests are conservative.

b) Since both histograms are similar, we will perform a two-sample t-test (either the equal variance or unequal variance version)

c) We conduct a two-sample t-test, but we write in the report that we probably made an error in the decision since the histogram of the math scores are bimodal.

d) We understand that the histogram of the math scores are bimodal, but if the difference between math scores of in-person and virtual students is normally distributed, a two-sample t-test is still suitable.

Question 10

Suppose the school board wants to determine if the level of math skills of students in virtual classes has decreased. Therefore, they collect the grade 4 math test scores for exactly the same students who took the test in grade 5 in a virtual class. What are the appropriate hypotheses in this case?

a) H₀: _d= 0vs. H_a: _d> 0where _dis (score of grade 5 -score of grade 4)

b) H₀: _grade4= _grade5vs. H_a: _grade4< _grade5

c) H₀: _grade4= _grade5vs. H_a: _grade4> _grade5

d) H₀: _d= 0vs. H_a: _d< 0where _dis (score of grade 5 - score of grade 4)

Question 11

Using the following information, calculate appropriate test statistic:

Smaple mean math score grade 4 = 83.8

Sample mean math score grade 5 = 83

sample standard deviation math scores grade 4 = 12.25

sample standard deviation math scores grade 5 = 15

sample standard deviation of the difference = 7.51

a) t= -0.476 with df= 39

b) t= -0.476 with df= 19

c) t=-2.13 with df=19

d) t=-0.106 with df= 19

e) t=-0.106 with df= 39

Question 12

At a 5% significance level, what is the appropriate confidence interval for testing the hypothesis from question 38 above?

a) (-3.70, )

b) (-4.30, 2.70)

c) (-, 2.10)

d) (0, 2.71)

e) (-3.70, 2.10)

Question 13

What happens if the histogram of the differences in math scores isNOTNormal?

a) We need to consider a Wilcoxon signed-rank test to do hypothesis testing.

b) The sample size is 20, which is large enough to assume normality. So, we should not have checked the histogram.

c) Any non-parametric test would be suitable as we have two samples and both Wilcoxon signed-rank and Wilcoxon rank-sum tests work with two samples.

d) We can conduct the paired t-test and then check the normal probability plot. Only then we can decide what to do.

e) We need to consider a Wilcoxon rank-sum test to do hypothesis testing.