The following information is for Questions 1 through 6.

The recommended daily allowance (RDA) of cobalamine (Vitamin B12) for growing teens is 2.4 μg (micrograms). My brother takes it upon himself to make sure that everyone gets the message. It is generally believed that growing teens are getting less than the RDA of 2.4 μg of cobalamine daily.

It is an open secret that Au & Associates FauxPharmaceutical (FAP) peddles dietary supplements around the globe. It is claimed by representatives of FAP that by taking their vitamin supplement extracted from Atlantic scrod, brand-named as Pillule-au-Sashimi, teens will have the RDA of cobalamine. Médecins Sans Frontières (MSF) has a strong contingent of volunteers in the People’s Republic of Banana. It is in this remote republic MSF is going to take on FAP for good.

MSF volunteers managed to collect with a 24-hour period blood samples of 10 randomly selected teens in the People’s Republic of Banana for cobalamine. The amounts of cobalamine (in μg) determined in these 10 randomly selected teens are given as follow:

1.85

2.35

1.87

1.90

1.37

2.35

2.55

2.28

1.95

2.49

Based on their global experience, MSF assumes that the population standard deviation of cobalamine in teens to be 0.56 μg.

Now, you are asked to weigh in on the dispute between Médecins Sans Frontières (MSF) and Au & Associates FauxPharamaceutical (FAP).

1. (3 points) Given the above information, what kind of hypothesis test will you conduct? z-test, t-test, Ω-test, χ2-test, Y-test, or F-test? Please explain.

2. (3 points) What will be the null hypothesis, the alternative hypothesis, and, hence, the “tailedness” of the test (left-tailed, right-tailed, or two-tailed)?

3. (10 points) What is be the corresponding test statistics?

4. (5 points) What is the corresponding p-value of the hypothesis test?

5. (15 points) What kind of conclusion can you draw from the hypothesis test you have just performed? Of course, representatives of FAP would like to have the conclusion skewed to their advantage. And so would the representatives from MSF. What would you do if you are representing FAP? But, if you are representing MSF, how would you present your argument?

6. (10 points) But, wait. What if MSF actually does not know the population standard deviation in this case, would you conduct your hypothesis test different? Just in case that you are going to perform the hypothesis different, what would you do instead?

The following information is for Question 7.

The tête-à-tête between MSF and FAP broke down, as anyone would have anticipated. They are going to court.

To prepare for the upcoming court case, FAP managed to get hold of the same 10 teens randomly selected by MSF, and performed its own testing on them. Their corresponding findings on the teens, in the same order as listed by MSF, are listed below.

2.45

2.85

2.87

2.32

1.98

2.51

1.75

1.98

2.03

2.89

It is understandable that FAP would have findings with a higher level of cobalamine in the teens. But it is important to authenticate FAP’s findings.

It is imperative to make sure that FAP actually have the same group of teens tested by MSF. His Honor, Judge Ig Sushi is presiding over this landmark case. Unexpectedly, Professor Au is deemed to be able to act impartially, and appointed by His Honor to be the amicus curiæ in this pending court case. (As a reward for his faithful service, Professor Au will be offered lifelong supply of bananas from the republic.)

After a few tasty bananas, Professor Au suggested to His Honor that the court should perform a statistical test to see if there is any difference in the means of the MSF group and the FAP group based on a court chosen significance level. Furthermore, Professor Au informed His Honor that it would be unlikely that either MSF or FAP actually knew the population standard deviation at all.

7. (16 points) Perform the statistical test recommended by Professor Au. What significance levels do you think MSF would prefer? On the other hand, what significance levels you think FAP would like to set?

8. (3 points) What is the probability of a Type I error?

9. (10 points) What if the RDA of cobalamine is actually not well established in the medical community? In fact, it may span a range of values in μg, 2.0, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6. What is the corresponding Type II error and power for each of the given assumed RDA value?

10. (5 points) Plot the corresponding power curve? What do we observed in this plot? Please explain.