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STT 315 Section 107 06/26/2017

Final Exam

– Throughout this exam, round numbers to three decimal digits

(unless the number contains less than 3 decimal digits to begin with).

– For any multiple choice question, circle the correct answer, DO NOT show any work.

– For any other question, write all the necessary work in the space provided making sure that

you are also following the guidelines of the part the question is in.

– For any question related to the binomial or normal random variable,

please write the calculator command that you are using before writing your final answer.

– If you need more space to write, first use the extra answer page (last page of the exam).

If you still need more space, attach pages

1

Part I:

1. In East Lansing 35% of the population votes for Democrats, 20% for Republicans and 45%

does not vote at all. In Lansing 26% of the population votes for Democrats, 29% for Republicans

and 45% does not vote at all. 10 individuals are randomly selected from each of the two cities

(10 from East Lansing, 10 from Lansing).

Let X be the number of people (out of the twenty people in the sample) who vote for Democrats,

let Y be the number of people (out of the twenty people in the sample) who vote for Republicans,

and let Z be the number of people (out of the twenty people in the sample) who do not vote. Then

(4 pts)

(A) All of X, Y, Z are binomial (B) Only X and Y are binomial

(C) Only Z is binomial (D) None of X, Y, Z is binomial

2. For any continuous random variable X, which of the following are true ? (4 pts)

(i) P(X > a) = P(X ≥ a) (ii) P(X < a) = P(X ≤ a) (iii) P(X = a) = 0

(A) Only (iii) (B) None of (i),(ii),(iii) (C) All of (i),(ii),(iii) (D) Only (i) and (ii)

3. There are 4 multiple choice questions on this exam, each with 4 possible answer choices. Suppose

that for each question, you pick an answer choice randomly. Let X be the number of multiple

choice questions that you answered correctly. Find the following quantities. (8 pts total, 2 pts each)

Simply write the appropriate formula(s) (or calculator command) and your final answer.

You do not need to explain anything.

(i) P(X = 2)

(ii) P(X < 4)

(iii) P(X ≥ 1)

(iv) The expected value and the standard deviation of X.

2

Part II:

– For each problem in this part, show all your work.

Start by specifying the sampling distribution (type of the distribution, its mean and standard deviation)

of the quantity that you are dealing with in the problem.

4. Suppose that the prices of plane tickets to China are normally distributed with mean $2500

and standard deviation $225. A firm wants to send 9 statisticians to China and has a budget of

$23400. What is the probability that the cost of the statisticians’ plane tickets would exceed the

budget ? Assume that the prices of the 9 tickets are independently distributed. (6 pts)

5. According to the American Dental Association, 60% of all dentists use nitrous oxide in their

practice. In a random sample of 75 dentists, let ˆp represent the proportion who use laughing gas

in practice. Find P(ˆp > .70). (4 pts)

3

Part III:

For any question in this part:

– DO NOT check whether the conditions to use a confidence interval hold

– write the formula of the confidence interval (or of the sample size) that you are supposed to find

– write the calculator command that you used to find your z-value or circle your t-value in the

T-table at the back of the exam

– for any confidence interval question, write your final answer either in the form ”(lower bound,

upper bound)” or in the form ”point estimate ± margin of error”, it is your choice.

6. It is known from past data that the standard deviation of the ages of online shoppers is 2.5

years and that the ages of online shoppers have an approximately normal distribution. In a recent

study, 16 online shoppers were randomly selected. The average age of shoppers in the sample was

23.79 years. Construct a 98% z-interval for the average age of the population of online shoppers.

(4 pts)

7. The Lincoln Tunnel (under the Hudson River) connects suburban New Jersey to midtown Manhattan.

Because of the substantial wait during rush hour, the Port Authority of New York and

New Jersey is considering raising the amount of the toll between 7 : 30 and 8 : 30 A.M. to encourage

more drivers to use the tunnel at an earlier or later time. Suppose the Port Authority

experiments with peak-hour pricing, increasing the toll from $4 to $7 during the rush hour peak.

On 81 randomly selected different workdays (during the period with the higher toll) at 8 : 30 A.M.

aerial photographs of the tunnel queues are taken and the number of vehicles counted. The average

and the standard deviation of the number of vehicles in line for those 81 days are 1174 and 160.68

respectively. Construct a 99% t-interval for the average number of vehicles in line during the rush

hour peak for the entire period with higher tolls. (4 pts)

4

8. The placebo effect describes the phenomenon of improvement in the condition of a patient taking

a placebo – a pill that looks and tastes real but contains no medically active chemicals. Physicians

at a clinic in La Jolla, California, gave what they thought were drugs to a random sample of 7, 000

asthma, ulcer, and herpes patients. Although the doctors later learned that the drugs were really

placebos, 70% of the patients reported an improved condition. Find a 95% confidence interval for

the proportion of all such patients who would report an improved condition despite being administered

a placebo. (4 pts)

9. Explain what the phrase ”90% confident” means when we interpret a 90% confidence interval

for µ. (4 pts)

(A) The probability that the given interval captures the sample mean is 0.90.

(B) In repeated sampling, 90% of similarly constructed intervals contain the value of the population

mean.

(C) 90% of the observations in the population fall within the bounds of the calculated interval.

(D) The probability that the population mean falls in the calculated interval is 0.90.

10. A company is interested in estimating µ, the mean number of days of sick leave taken by

all its employees. Assume that a reasonable guess for the standard deviation of the number of days

of sick leave taken by the company’s employees is 10 days. How many personnel files would the

company’s statistician have to select in order to estimate µ to within 2 days with a 90% confidence

interval ? (4 pts)

5

11. Recently, a case of salmonella (bacterial) poisoning was traced to a particular brand of ice

cream bar, and the manufacturer removed the bars from the market. Despite their response, many

consumers refused to purchase any brand of ice cream bars for some period of time after the event

(McClave, personal consulting). It is now 1 year after the outbreak of food poisoning was traced

to ice cream bars. The manufacturer wishes to estimate the proportion who still will not purchase

bars to within .02 using a 95% confidence interval. (8 pts total, 4 pts each)

(i) How many consumers should be sampled if there is no prior guess about the proportion ?

(ii) How many consumers should be sampled if it is known that a reasonable guess for the proportion

is .10 ?

6

Part IV:

12. A company claims that the mean age of retirement of their employees is higher than 70.

The employees ask the company to convince them about the truth of the claim.

Fill in the appropriate hypotheses below. (2 pts)

H0 : Ha :

13. A company claims that the proportion of their employees who retire before turning 65 is

less than 30%. The employees ask the company to convince them about the truth of the claim.

Fill in the appropriate hypotheses below. (2 pts)

H0 : Ha :

14. Which of the following statements are true ? (6 pts)

In a testing problem if the null hypothesis is

(i) rejected at 5% significance level, then it will also be rejected at 1% significance level.

(ii) rejected at 1% significance level, then it will also be rejected at 5% significance level.

(iii )accepted at 5% significance level, then it will also be accepted 1% significance level.

(iv) accepted at 1% significance level, then it will also be accepted at 5% significance level.

(A) (i) and (iii) (B) (i) and (iv) (C) (ii) and (iii) (D) (ii) and (iv)

For Problems 15 through 17, use the calculator only as a computational tool.

YOU MAY use the Z-Test, T-Test, or 1-PropZTest functions in the calculator to solve the entire

problem.

Any time you have to compute a test statistic, write the formula that you are using with the appropriate

values from the problem.

Any time you have to find a rejection region or a p-value, draw a graph and shade the area of

interest.

7

15. Last year, the ages of online shoppers were approximately normally distributed with mean 25

years and standard deviation 2.5 years. A sample of 16 online shoppers was randomly selected

during the current year. The average age of shoppers in the sample was 23.79 years . Assume that

the shape of the distribution and the standard deviation of ages in the population did not change

since last year. It is desired to test if the average age of online shoppers has changed this year.

We want to test H0 : µ = 25 against Ha : µ 6= 25. (13 pts total)

(i) Are the necessary conditions for a valid test of hypothesis for µ satisfied in this problem ?

Explain. (3 pts)

(ii) Should we use a z-statistic or a t-statistic to carry out the test ? Explain. (1 pt)

(iii) Find the value of the test statistic. (2 pts)

8

(iv) Find the rejection region for a 2% level of significance (α = .02). (3 pts)

(v) State the appropriate conclusion at 2% level of significance. (2 pts)

(vi) Suppose that it turns out the true mean this year is µ = 23.75.

What can you say about the decision you made in the previous question ? (2 pts)

9

16. The Lincoln Tunnel (under the Hudson River) connects suburban New Jersey to midtown Manhattan.

On Mondays at 8 : 30 A.M., the mean number of cars waiting in line to pay the Lincoln

Tunnel toll is 1, 220. Because of the substantial wait during rush hour, the Port Authority of New

York and New Jersey is considering raising the amount of the toll between 7 : 30 and 8 : 30 A.M.

to encourage more drivers to use the tunnel at an earlier or later time. Suppose the Port Authority

experiments with peak-hour pricing for a year, increasing the toll from $4 to $7 during the rush

hour peak. On 81 randomly selected different workdays (during the period with the higher toll) at

8 : 30 A.M. aerial photographs of the tunnel queues are taken and the number of vehicles counted.

The average and the standard deviation of the number of vehicles in line for those 81 days are

1174 and 160.68 respectively. We would like to determine whether peak-hour pricing succeeded in

reducing the average number of vehicles attempting to use the Lincoln Tunnel.

We want to test H0 : µ = 1220 against Ha : µ < 1220. (13 pts total)

(i) Are the necessary conditions for a valid test of hypothesis for µ satisfied in this problem ?

Explain. (3 pts)

(ii) Should we use a z-statistic or a t-statistic to carry out the test ? Explain. (1 pt)

(iii) Find the value of the test statistic. (2 pts)

10

(iv) Find the p-value corresponding to the test statistic. (3 pts)

(v) State the appropriate conclusion at 1% level of significance. (2 pts)

(vi) Suppose that it turns out the true mean is still µ = 1220.

What can you say about the decision you made in the previous question ? (2 pts)

11

17. The placebo effect describes the phenomenon of improvement in the condition of a patient

taking a placebo – a pill that looks and tastes real but contains no medically active chemicals.

Physicians at a clinic in La Jolla, California, gave what they thought were drugs to a random

sample of 7, 000 asthma, ulcer, and herpes patients. Although the doctors later learned that the

drugs were really placebos, 70% of the patients reported an improved condition. If the placebo

was ineffective, the probability of a patient’s condition improving would be 0.5. We would like to

determine if there is enough evidence to conclude that the placebo is effective.

We want to test H0 : p = 0.5 against Ha : p > 0.5. (10 pts total)

(i) Are the necessary conditions for a valid test of hypothesis for p satisfied in this problem ?

Explain. (3 pts)

(ii) Find the value of the test statistic. (2 pts)

(iii) Find the p-value corresponding to the test statistic. (3 pts)

(iv) State the appropriate conclusion at 5% level of significance. (2 pts)

12

18. School administrators collect data on students attending the school. Which of the following variables

is qualitative?

A. Gross income of the parents or Guardian B. G.P.A

C. Age of the student D. whether the student has taken the SAT

19. At the Paris airshow, suppose that the following 65 aircraft sales resulted. Suppose that a sale is

chosen at random from the 65 sales.

Manufacturer

Purchaser Boeing Airbus

Finnair 0 12

TAM (Brazil) 8 5

Northwest Airlines 12 8

British Ariways 10 10

a) What is the probability that the sale was either by Boeing or to TAM?

A. 43/65 B. 30/65 C. 8/65 D. 35/65

b) Given that a sale is by Boeing, find the probability that it is to British Airways.

A. 10/65 B. ½ C. 10/30 D. 10/35

c) Given that the sale is to Finnair, what is the probability that it is by Airbus?

A. 1 B. 0 C. 12/35 D. 12/65

20. Consider the model where the continuous random variable X is uniformly distributed U(10,50).

a) What is the probability that X is more than 40 i.e. P(X≥ 40)?

A. 0.003 B. 0.897 C. 0.250 D. 0.750

b) What is the expected value and the standard deviation of X?

A. 30, 11.55 B. 30, 133.33 C. 11.55, 133.33 D. 30, 1.566

21. A device has three components C1, C2 and C3. The device works as long as at least one of the

components is functional. The probabilities of the components being functional are 0.7, 0.9 and 0.6

respectively. Assume independence.

a) What is the probability that C1 is functional but C2 and C3 are not functional?

A. 0.504 B. 0.054 C. 0.006 D. 0.028

b) What is the probability that the device completely fails?

A. 0.378 B. 0.012 C. 0.639 D. 0.452

6. Five True or False questions are given to a student. Suppose that the student selects his answers by 5

independent tosses of a fair coin with say, H=True, T=False. (This is a model for guessing)

a) What is the probability that the student gets all the questions wrong?

A. 0.031 B. 0.969 C. 0.5 D. 0.25

b) What is the probability that the student gets at least one correct answer?

A. 0.339 B. 0.471 C. 0.969 D. 0.225

22. Below is a stem and leaf display of the ages of n=42 children ranging in ages from 1 to 38 months.

0|1 1 2 2 3 4 5 5 5 6 7 7 7 8 8 9 9

1|0 0 1 2 3 4 4 4 5 5 6 6 8

2|0 1 3 4 5 5 6 7

3|1 2 3 8

a) What is the median age?

A. 22 months B. 12.5 months C. 20 months D. 15 months

b) What is the IQR of the ages?

A. 7 months B. 21 months C. 6 months D. 14 months

23. U.S airlines average about 4.5 fatalities per month. Assume the probability distribution for X, the

number of fatalities per month, can be approximated by a Poisson probability distribution.

a) What is the probability that no fatalities will occur during any given month?

A. 0.122 B. 0.011 C. 0.639 D. 0.995

b) What is the probability that either 2 or 3 fatalities occur during a month?

A. 0.033 B. 0.174 C. 0.169 D. 0.281

c) What is the probability that at least 4 fatalities occur during a month?

A. 0.658 B.0.468 C. 0.199 D. 0.008

24. Suppose a large labor union wishes to estimate the mean number of hours per month a union member

is absent from work. The union decides to sample 468 of its members at random and monitor the working

time of each of them for 1 month. At the end of the month, the total number of hours absent from work is

recorded for each employee. Which of the following should be used to estimate the parameter of interest

for this problem?

A. A small sample confidence interval for p. B. A small sample confidence interval for μ (T-Int)

C. A large sample confidence interval for p. D. A large sample confidence interval for μ (Z-Int)

25. A chemical plant has an emergency alarm system for a given batch reaction. When an emergency

situation exists, the alarm sounds with probability 0.9. When an emergency situation does not exist, the

alarm system sounds with a probability 0.05. Based on investigations and records, an emergency situation

with the batch reaction is a rare event, it has a probability 0.004.

a) What is the probability that the alarm sounds for a batch reaction?

A. 0.004 B. 0.033 C. 0.053 D. 0.166

b) Given that the alarm has just sounded, what is the probability that an emergency situation exists?

A. 0.067 B. 0.500 C. 0.160 D. 0.022

c) Given that the alarm has not sounded, what is the probability that an emergency situation exists?

A. 0.076 B. 0.153 C. 0.0004 D. 0.0001

26. Consider two random variables X and Y. The expected value of X is 4.5 and standard deviation is 1.5.

The expected value of Y is 10.6 and standard deviation is 3.

a) What is the value of E (-5X+3)?

A. 23.5 B. -19.5 C. -63.2 D. -22.5

b) What is the value of S.D (-2Y+1.5)?

A. 6 B. -12 C. 3 D. 36

c) What is the value of E (4Y-2.5X)?

A. 30.21 B. 0.36 C. 67.5 D. 31.15

27. Early in 2007 Consumer Reports published the results of an extensive investigation of broiler

chickens purchased from food stores in 23 states. Tests for bacteria in the meat showed that 81% of the

chickens were contaminated with campylobacter, 15% with salmonella and 13% with both.

a) What’s the probability that a tested chicken was not contaminated with either kind of bacteria?

A. 0.17 B. 0.83 C. 0.96 D. 0.04

b) What’s the probability that a tested chicken was contaminated with salmonella alone?

A. 0.15 B. 0.02 C. 0.13 D. 0

28. Below is a boxplot of 18 measurements.

a) Approximately what percentage of measurements is above 3?

A. 50% B. 25% C. 75% D. 95%

b) By looking at the distribution of the measurements, which among the following options is correct?

A. Mean = Median B. Mean > Median C. Mean <

Median

c) If a measurement that was originally evaluated to be 18 was replaced by a measurement with value 4,

would you expect the median of the data to,

A. Shift to the right or remain the same B. Shift to the left or remain the same

C. just remain the same D. Shift to the left

29. Consider the population with the following probability distribution for a characteristic X.

X 1 2 3 4 5

P(X) 0.2 0.3 0.2 0.2 0.1

a) P(1<X≤ 3)= A. 0.5 B. 0.7 C. 0.3 D. 0.63

b) E(X)= μ and SD(X)=σ , what is μ and σ ?

A. 2.5, 1.36 B. 3.5,1.440 C. 2.7, 1.269 D. 2.7, 0.366

A random sample of size n=2, is to be drawn,

c) P(X̅≤ 3)= A. 0.32 B. 0.63 C. 0.77 D. 0.45

d) What are E(X̅) and SD(X̅ )?

A. 2.7, 0.897 B. 2.7,1.269 C. 2.5, 0.962 D. 2.5, 0.311

30. A student scores one standard deviation above the mean on an exam given to 1000 students. The

scores are mound shaped and symmetrically distributed. The student then scored above how many

students approximately?

A. 840 B. 900 C. 925 D. 997

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