For questions 1-6, refer to the following information: Researchers conducted a study to investigate the factors affecting the progression of pneumonia in novel coronavirus disease 2019 (COVID-19) patients who were admitted to three tertiary hospitals in Wuhan between December 30, 2019 and January 15, 2020. They collected individual data, laboratory indices, imaging characteristics, and clinical data, including albumin levels (g/L). Albumin is a protein produced by your liver that helps to keep fluid in your bloodstream, so it does not leak into other tissues. It also carries hormones, vitamins, and enzymes throughout your body. Low albumin levels can indicate a problem with your liver or kidneys. Based on clinical typing results, the patients were divided into a progression group (n=11) or an improvement/stabilization group (n=67). The mean albumin level in the progression group was 36.62 g/L with a standard deviation of 6.60 g/L. The mean albumin level in the improvement/stabilization group was 41.27 g/L with a standard deviation of 4.55 g/L. Using a 0.05 significance level, you need to test the claim that patients in the progression group had significantly lower albumin levels than patients in the improvement/stabilization group. To do so, assume the population variances are equal and answer each of the following questions.

1. What is the null hypothesis? (1 point)

1. x1x2
2. x1
3. 12
4. 1-2<0
5. 1-20

1. What is the alternative hypothesis? (1 point)

1. x1x2
2. x1
3. 12
4. 1-2<0
5. 1-20

1. What is the value of the test statistic you should use to test the given hypothesis? (2 points)

1. -1.58
2. -1.65
3. -2.25
4. -2.94
5. -4.65

1. If you were instead told to assume that the population variances are unequal, what value would you use for the denominator of the test statistic, i.e. the standard error of your relevant statistic? (1 point)

1. 1.58
2. 2.07
3. 2.50
4. 4.28
5. 4.87

1. What is the p-value (corresponding to your test statistic in problem 3)? (1 point)

1. 0.9978
2. 0.9956
3. 0.05
4. 0.002
5. 0.0011

1. What conclusion should you make based on the given data? (1 point)

1. Fail to reject H0 and conclude there is not sufficient evidence to support the claim that patients in the progression group have a lower mean albumin level than patients in the improvement/stabilization group.
2. Fail to reject H0 and conclude there is sufficient evidence to support the claim that patients in the progression group have a lower mean albumin level than patients in the improvement/stabilization group.
3. Accept H0 and conclude there is sufficient evidence to support the claim that patients in the progression group have a lower mean albumin level than patients in the improvement/stabilization group.
4. Reject H0 and conclude there is not sufficient evidence to support the claim that patients in the progression group have a lower mean albumin level than patients in the improvement/stabilization group.
5. Reject H0 and conclude there is sufficient evidence to support the claim that patients in the progression group have a lower mean albumin level than patients in the improvement/stabilization group.

For questions 7-12, refer to the following information: Researchers conducted a study to evaluate which of two medications, Drug A or Drug B, is more effective in treating headaches. Each patient was administered Drug A for one headache and Drug B for a different headache. The amount of time to pain relief was recorded in the table below. The sample standard deviation of the differences between the relief time of Drug A and the relief time of Drug B is 2.1213. Using a 0.05 significance level, you need to test the claim that the relief time for Drug A is different than the relief time for drug B. To do so, answer each of the following questions.

Drug A Relief Time

20

40

30

45

19

27

32

26

Drug B Relief Time

18

36

30

46

15

22

29

25

1. What is the most appropriate statistical test for the given claim? (0.5 points)

1. Two-sample z-test
2. Wilcoxon rank-sum test
3. Two-sample (pooled variance) t-test
4. Two-sample (unpooled variance) t-test
5. Paired t-test

1. What is the null hypothesis H0? (1 point)

1. d=0
2. d0
3. 1-2=0
4. 1-20
5. x1-x2=0
6. x1-x20

1. What is the alternative hypothesis H1? (1 point)

1. d=0
2. d0
3. 1-2=0
4. 1-20
5. x1-x2=0
6. x1-x20

1. What critical value(s) should you use to test the given hypothesis? (1 point)

1. 1.860
2. 1.895
3. 1.960
4. 2.306
5. 2.365

1. What is the value of the test statistic you should use to test the given hypothesis? (2 points)

1. 0.75
2. 2.25
3. 2.81
4. 3.00
5. 8.49

1. What conclusion should you make based on the given data? (1 point)

1. Reject H0 and conclude there is not sufficient evidence to support the claim that the relief time for Drug A is different than the relief time for drug B.
2. Reject H0 and conclude there is sufficient evidence to support the claim that the relief time for Drug A is different than the relief time for drug B.
3. Accept H0 and conclude there is sufficient evidence to support the claim that the relief time for Drug A is different than the relief time for drug B.
4. Fail to reject H0 and conclude there is not sufficient evidence to support the claim that the relief time for Drug A is different than the relief time for drug B.
5. Fail to reject H0 and conclude there is sufficient evidence to support the claim that the relief time for Drug A is different than the relief time for drug B.

For questions 13-18, refer to the following information: Mancia et al. (2020) conducted a case-control study in the Lombardy region of Italy to examine the association between the use of angiotensin-converting-enzyme (ACE) inhibitors and the odds of coronavirus disease 2019 (COVID-19). There were 1,502 COVID-19 cases among the 8,071 patients who used ACE inhibitors. There were 4,770 COVID-19 cases among the 28,960 patients who did not use ACE inhibitors. Using a 0.05 significance level, you need to test the claim that the proportion of COVID-19 patients among those who used ACE inhibitors (group 1) is higher than the proportion of COVID-19 patients among those who did not use ACE inhibitors (group 2). To do so, answer each of the following questions.

1. What proportion of ACE inhibitor users were not COVID-19 cases? (0.5 points)

1. 0.14
2. 0.16
3. 0.19
4. 0.81
5. 0.84

1. What is the null hypothesis H0? (1 point)

7. None of the above because we should be making our claim about and.

1. What is the alternative hypothesis H1? (1 point)

7. None of the above because we should be making our claim about and.

1. What is the value of the test statistic you should use to test the given hypothesis? (2 points)

1. 0.21
2. 0.29
3. 1.65
4. 2.13
5. 4.53

1. What is the p-value (corresponding to your test statistic)? (1 point)

1. >0.9999
2. 0.95
3. 0.05
4. <0.0001
5. Cannot be determined from the given information

1. What conclusion should you make based on the given data? (1 point)

1. Fail to reject H0 and conclude there is sufficient evidence to support the claim that the proportion of COVD-19 cases is higher among those who used ACE inhibitors than those who did not use ACE inhibitors.
2. Fail to reject H0 and conclude there is not sufficient evidence to support the claim that the proportion of COVD-19 cases is higher among those who used ACE inhibitors than those who did not use ACE inhibitors.
3. Accept H0 and conclude there is sufficient evidence to support the claim that the proportion of COVD-19 cases is higher among those who used ACE inhibitors than those who did not use ACE inhibitors.
4. Reject H0 and conclude there is sufficient evidence to support the claim that the proportion of COVD-19 cases is higher among those who used ACE inhibitors than those who did not use ACE inhibitors.
5. Reject H0 and conclude there is not sufficient evidence to support the claim that the proportion of COVD-19 cases is higher among those who used ACE inhibitors than those who did not use ACE inhibitors.

I have attached pictures as well.