1. A government agency is receiving complaints from customers that a restaurant in city has been selling hamburgers which are underweight. The agency would like to do investigation by testing following hypotheses. 

Restaurant is doing what it is supposed to and not selling underweight hamburgers:

The restaurant is doing not what it is supposed to and is selling underweight hamburgers.

Please answer following: 

(i) If Type I error occurs then what could be the conclusion?

(ii) If Type II error occurs then what is the resulting conclusion?

(iii) What would be the more serious error for restaurant? 

2. A scientist believes that a population mean is exceeding to 250. He decides to take a random sample of 100 measurements. 

The sample comes out to be 259 and Standard Deviation is 40. The value of α= 0.05. 

Find following:

(a) Construct a null and alternative hypothesis.

(b) Find significance level and explain what does sit mean

(c) Calculate Test statistic

(d) Find out rejection region

(e) What conclusion could be drawn?

3. In this recession, yours truly, CEO of the Outrageous Products Enterprise, would like to make extra money to support my frequent filet-mignon-and- double-lobster-tail dinner habit. 

A promising enterprise is to mass-produce tourmaline wedding rings for brides. Based on my diligent research, I have found out that women's ring size normally distributed with a mean of 6.0, and a standard deviation of 1.0. I am going to order 5000 tourmaline wedding rings from my reliable Siberian source. They will manufacture ring size from 4.0, 4.5, 5.0, 5.5, 6.0, 6.5, 7.0, 7.5, 8.0, 8.5, 9.0, and 9.5. 

How many wedding rings should I order for each of the ring size should I order 5000 rings altogether? (Note: It is natural to assume that if your ring size falls between two of the above standard manufacturing size, you will take the bigger of the two.)

4. A company manufactures microwave for popcorn and claims that only 2.5% of the popcorn failed to pop. Another company who is competitor believes that percentage could be much higher. So this second company does tests with 3000 kernels and finds that at least 50 of them failed to pop. Do you think that these results are sufficient to verify the beliefs of the competitor? Try to do the tests at 0.01 significance level.

5. It is commonly accepted that the mean temperature of human is 98.6oF. 

Yours truly has nothing better to do but measured the temperatures of 26 colleagues 1 to 4 times daily to get a total of 123 measurements. 

The collected data yielded a sample mean of 98.4oF and a sample standard deviation of 0.7oF. Is the mean temperature of his colleagues less than 98.6oF at the 0.01 significance level? 

Justify your answer with the proper statistics.

The following information is for Questions 6  through 11. 

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.

6. Given the above information, what kind of hypothesis test will you conduct? Should it be α-test, z-test, t-test, χ2-test, F-test, or Ω-test? Please explain.

7. What will be the null hypothesis, the alternative hypothesis, and, hence, the "tailedness" of the test (left-tailed, right-tailed, or two-tailed)? 

8. What is be the corresponding test statistics?

9. What is the corresponding p-value of the hypothesis test? 

10. 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?

11. 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?