The recommended daily allowance (RDA) of cobalamine (Vitamin B12) for growing teens is 2.4 µg (micrograms). It is generally believed that growing teens are getting less than the RDA of 2.4 µg of cobalamine daily.
A not-to-be-named Pharmaceutical (ntbnP) peddles dietary supplements around the country. It is claimed by ntbnP representatives that by taking their vitamin supplement, teens will have the RDA of cobalamine. FDA is going to take on ntbnP to show that the supplement comes short of providing teens with the recommended RDA.
FDA managed to collect with a 24-hour period blood sample of 10 randomly selected teens around the country. 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 national experience, FDA assumes that the the population standard deviation of cobalamine in teens to be 0.56 µg.
Now, you are asked to weigh in on the dispute between FDA and ntbnP.
a. Given the above information, what kind of hypothesis test will you conduct? z-test, t-test, χ2-test, F-test, or Ω-test? Please explain.
b. What will be the null hypothesis, the alternative hypothesis, and, hence, the "tailedness" of the test (left-tailed, right-tailed, or two-tailed)?
c. What is be the corresponding test statistics?
d. What is the corresponding p-value of the hypothesis test?
e. What kind of conclusion can you draw from the hypothesis test you have just performed? Of course, representatives of ntbnP would like to have the conclusion skewed to their advantage. And so would the officials from FDA. What would you do if you are representing ntbnP? But, if you are representing FDA, how would you present your argument?
f. But, wait. What if FDA 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?