1. (5 points) Most of us love Luzon mangoes, but hate buying those that are picked too early. Unfortunately, by waiting until the mangos are almost ripe to pick carries a risk of having 15% of the picked rot upon arrival at the packing facility. If the packing process is all done by machines without human inspection to pick out any rotten mangos, what would be the probability of having at most 2 rotten mangos packed in a box of 12?
2. (5 points) We have 7 boys and 3 girls in our church choir. There is an upcoming concert in the local town hall. Unfortunately, we can only have 5 youths in this performance. This performance team of 5 has to be picked randomly from the crew of 7 boys and 3 girls.
a. What is the probability that all 3 girls are picked in this team of 5?
b. What is the probability that none of the girls are picked in this team of 5?
c. What is the probability that 2 of the girls are picked in this team of 5?
3. (10 points) In this economically challenging time, yours truly, CEO of the Outrageous Products Enterprise, would like to make extra money to support his frequent filet-mignon-anddouble-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. (5 points) A soda company want to stimulate sales in this economic climate by giving customers a chance to win a small prize for ever bottle of soda they buy. There is a 20% chance that a customer will find a picture of a dancing banana ( ) at the bottom of the cap upon opening up a bottle of soda. The customer can then redeem that bottle cap with this picture for a small prize. Now, if I buy a 6-pack of soda, what is the probability that I will win something, i.e., at least winning a single small prize?
5. (10 points) Below is a summary of the Quiz 1 for two sections of STAT 225 last spring. The questions and possible maximum scores are different in these two sections. We notice that Student A4 in Section A and Student B2 in Section B have the same numerical score. Section A Student Score Section B Student Score A1 70 B1 15 A2 42 B2 61 A3 53 B3 48 A4 61 B4 90 A5 22 B5 85 A6 87 B6 73 A7 59 B7 48 —– —— B8 39 How do these two students stand relative to their own classes? And, hence, which student performed better? Explain your answer.
6. (5 points) An amusement park is considering the construction of an artificial cave to attract visitors. The proposed cave can only accommodate 36 visitors at one time. In order to give everyone a realistic feeling of the cave experience, the entire length of the cave would be chosen such that guests can barely stand upright for 98% of the all the visitors. The mean height of American men is 70 inches with a standard deviation of 2.5 inches. An amusement park consultant proposed a height of the cave based on the 36-guest-at-a-time capacity. Construction will commence very soon. The park CEO has a second thought at the last minute, and asks yours truly if the proposed height is appropriate. What would be the proposed height of the amusement park consultant? And do you think that it is a good recommendation? If not, what should be the appropriate height? Why?
7. (5 points) A department store manager has decided that dress code is necessary for team coherence. Team members are required to wear either blue shirts or red shirts. There are 9 men and 7 women in the team. On a particular day, 5 men wore blue shirts and 4 other wore red shirts, whereas 4 women wore blue shirts and 3 others wore red shirt. Apply the Addition Rule to determine the probability of finding men or blue shirts in the team.
Please refer to the following information for Question 8.
It is an open secret that airlines overbook flights, but we have just learned that bookstores underbook (I might have invented this new term.) textbooks in the good old days that we had to purchase textbooks. To make a long story short, once upon a time, our UMUC designated virtual bookstore, MBS Direct, routinely, as a matter of business practice, orders less textbooks than the amount requested by UMUC’s Registrar’s Office. That is what I have figured out……. Simply put, MBS Direct has to “eat” the books if they are not sold. Do you want to eat the books? You may want to cook the books before you eat them! Oops, I hope there is no account major in this class? OK, let us cut to the chase….. MBS Direct believes that only 85% of our registered students will stay registered in a class long enough to purchase the required textbook. Let’s pick on our STAT 200 students. According to the Registrar’s Office, we have 600 students enrolled in STAT 200 this spring 2014.
8. (10 points) Suppose you are the CEO of MBS Direct, and you want to perform a probability analysis. What would be the number of STAT 200 textbook bundles you would order so that you stay below 5% probability of having to back-order from Pearson Custom Publishing? (Note: Our Provost would be very angry when she hears that textbook bundles have to be backordered. In any case, we no longer need the service of MBS Direct as we are moving to 100% to free eResources or OLI. Auf Wiedersehen, MBS Direct……) IMPORTANT: Yes, you may use technology for tacking Question 8 in this quiz.
Please refer to 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).
9. (2 points) Given the above information, what kind of hypothesis test will you conduct? ztest, t-test, Ω-test, χ 2 -test, Y-test, or F-test? Please explain.
10. (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)?
11. (5 points) What is be the corresponding test statistics?
12. (5 points) What is the corresponding p-value of the hypothesis test?
13. (5 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?
14. (5 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 differently, what would you do instead? Perform the necessary test.
Please refer to the following information for Question 15.
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.
15. (10 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?
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