Pamantasan ng Lungsod ng MaynilaIntramuros, Manila

College of Engineering and TechnologyEngineering Science Department

Engineering Probability and StatisticsTAKE HOME QUIZ

NAME: _________________________________________ COURSE: _______________ NICKNAME: __________________DATE: ____________________ INSTRUCTOR: _____________________

INSTRUCTIONS: Use separate answer sheet to answer the following. BOX ONLY THOSE ANSWERS NEEDED. Unboxed answers are considered incorrect. Express your answer in fraction if possible. Otherwise, use four (4) decimal places.

1. Given a standard normal distribution with µ = 30 and σ = 6, finda) the normal curve area to the right of x = 17.b) the normal curve area to the left of x = 22.c) the normal curve area between x = 32 and x = 41.d) the value of x that has 80% of the normal curve area to the left.e) the two values of x that contain the middle 75% of the normal curve area.

2. Find the value of z if the area under a standard normal curvea) to the right of z is 0. 3602.b) to the left of z is 0.1311.c) between 0 and z, with z>0, is 0.4338.d) between –z and z, with z>0 is 0.9500.

3. The daily amount of coffee, in liters, dispensed by a machine located in an airport lobby is a random variable X having a continuous uniform distribution with A = 7 and B = 10. Find the probability that on a given day the amount of coffee dispensed by this machine will be

a) at most 8.8 liters.b) more than 7.4 liters but less than 9.5 liters.c) at least 8.5 liters.

4. A bus arrives every 20 minutes at a bus stop. It is assumed that the waiting time for a particular individual is a random variable with a continuous uniform distribution.

a) What is the probability that the individual waits more than 10 minutes?b) What is the probability that the individual waits between 3 and 8 minutes?

5. The IQ’s of 600 applicants to a certain college are approximately normally distributed with a mean of 115 and a standard deviation of 12. If the college requires an IQ of at least 95, how many of these students will be rejected on this basis of IQ, regardless of their other qualifications? Note that IQs are recorded to the nearest integers.

6. A coin is tossed 500 times. Use the normal curve approximation to find the probability of obtaining a) between 185 and 210 heads inclusive.b) exactly 205 heads.c) fewer than 176 or more than 227 heads.

7. Statistics released by the National Highway Traffic Safety Administration and the National Safety Council show that on an average weekend night, 1 out of every 10 drivers on the road is drunk. If 400 drivers are randomly checked next Saturday night, what is the probability that the number of drunk drivers will be

a) less than 32?b) more than 49?c) at least 35 but less than 47?

8. A soft-drink machine is regulated so that it discharges an average of 200 milliliters per cup. If the amount of drink is normally distributed with a standard deviation equal to 15 milliliters,

a) what fraction of the cups will contain more than 224 milliliters?b) what is the probability that a cup contains between 191 and 209 milliliters?c) how many cups will probability overflow if 230-milliliter cups are used for the next 1000 drinks?d) below what value do we get the smallest 25% of the drinks?

9. A fabric manufacturer believes that the proportion of orders for raw material arriving late is p = 0.6 If a random sample of 10 orders shows that 3 or fewer arrived late, the hypothesis that p = 0.6 should be rejected in favor of the alternative p < 0.6. Use binomial distribution.

a) Find the probability of committing a type I error if the true proportion is p = 0.6.b) Find the probability of committing a type II error for the alternatives p = 0.3, p = 0.4, and p = 0.5.

10. A dry cleaning establishment claims that a new spot remover will remove more than 70% of the sports to which it is applied. To check this claim, the spot remover will be used on 12 spots chosen at random. If fewer than 11 of the spots are removed, we shall not reject the null hypothesis that p = 0.7; otherwise, we conclude that p > 0.7.

a) Evaluate α, assuming that p = 0.7.b) Evaluate β for the alternative p = 0.9.

11. A random sample of 400 voters in a certain city are asked if they favor an additional 4% gasoline sales tax to provide baldy needed revenues for street repairs. If more than 220 but fewer than 260 favor the sales tax, we shall conclude that 60% of the voters are for it.

a) Find the probability of committing a type I error if 60% of the voters favor the increased tax.b) What is the probability of committing a type II error using this test procedure if actually only 48% of the voters are in favor of the additional gasoline tax?

12. According to a dietary study, high sodium intake may be related to ulcers, stomach cancer, and migraine headaches. The human requirement for salt is only 220 milligrams per day, which is surpassed in most single servings of ready-to-eat cereals. If a random sample of 20 similar servings of a certain cereal has a mean sodium content of 244 milligrams and a standard deviation of 24.5 milligrams, does this suggest at the 0.05 level of significance that the average sodium content for a single serving of such cereal is greater than 220 milligrams? Assume the distribution of sodium contents to be normal.

13. A random sample of 64 bags of white cheddar popcorn weighed, on average, 5.23 ounces with a standard deviation of 0.24 ounce. Test the hypothesis that µ = 5.5 ounces against the alternative hypothesis, µ < 5.5 ounces, at the 0.05 level of significance.

14. From long experience with a process for manufacturing an alcoholic beverage it is known that the yield is normally distributed with a mean of 500 and a standard deviation of 96 units. For a modified process the yield is 535 units for a sample o size 50. At α = 0.05 does the modified process increase the yield?

15. In a report prepared by the Economic Research Department of a major bank, the Department manager maintains that the average annual family income on Metropolis is $48,432. What do you conclude about the validity of the report if a random sample of 300 families shows an average income of $48,574 with a standard deviation of 1000? Use alternative hypothesis µ >$48,432 at α = 0.05.

16. The employees of the local manufacturing plant normally pledged an average of $20 to the United Fund every year. This year, a random sample of employees donated the following amount in dollars: 10, 40, 30, 5, 25, 50, 35, 30, 5, 15, 30, 50, 5, 30, 10, 25, 45, and 10. With this donation, does it imply that the salary of the employees increases? Use α = 0.10.

Additional Instructions:

1. This take home exam is to be passed on October 6, 2012 (Saturday) at exactly 8:00 am. A deduction of 1 pt for every minute late will be applied.

2. Grading system will be 70% for the content and 30% for the Engineering Lettering (0, 15, 30).3. Passing of the take home exam should be done individually. Upon submission, you are to sign as a proof

that you submit your take home exam. 4. Use this page as your front page. 5. Staple your papers properly. Any detached papers after submission won’t be checked. 6. Succeeding answer sheet papers should follow Problem Set’s margin format. 7. Partial points will not be given for this exam.