Sample Final[1]

UNIVERSITY OF MACAU FACULTY OF SCIENCE AND TECHNOLOGY MATH 111 Probability and Statistics SAMPLE QUESTIONS III 1. The output voltage of a power supply is assumed to be normally distributed. Sixteen observations taken at random on voltage are shown here. 10.35 9.30 11.65 12.00 11.54 9.95 10.44 9.25 a) Test the hypothesis that

10.00 11.25 10.28 9.38 the mean

9.96 9.58 8.37 10.85 voltage equals 12V against a two-sided

alternative using α = 0.05. b) Construct a 95% two-sided confidence interval on µ . c) Test the hypothesis that σ 2 = 11 using α = 0.05. d) Construct a 95% two-sided confidence interval on σ . e) Construct a 95% upper confidence interval on σ . State your conclusions or the meaning of the confidence intervals in words for the above questions.

2. The manufacturer of a synthetic fiber claims the mean tensile strength is 50 psi. A random sample of 16 fiber specimens is selected, and the sample mean strength is 49.86 psi and the sample standard deviation is 1.66 psi. Assuming that tensile strength is approximately normally distributed, a) construct a 95% two-sided confidence interval on the mean tensile strength. Does the data support the manufacturer’s claim? b) Construct a 95% lower confidence interval on the mean tensile strength. c) Construct a 95% upper confidence interval on the standard deviation of tensile strength.

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3. Two quality control technicians measured the surface finish of a metal part, obtaining the data shown. Assume that the measurements are normally distributed. Technician 1 1.45 1.37 1.21 1.54 1.48 1.29 1.34

Technician 2 1.54 1.41 1.56 1.37 1.20 1.31 1.27 1.35

a) Test the hypothesis that the mean surface finish measurements made by the two technicians are equal. Use α = 0.05, and assume equal variances. b) Assuming that the variances are equal, construct a 95% confidence interval on the mean difference in surface- finish measurements. c) Test the hypothesis that the variances of the measurements made by the two technicians are equal. Use α = 0.05. d) Construct a 95% confidence interval estimate of the ratio of the variances of technician measurement error. e) Construct a 95% confidence interval on the varia nce of measurement error for technician 2. State your conclusions or the meaning of the confidence intervals in words for the above questions. 4. TIME, April 18, 1994, reported the results of a telephone poll of 800 adult Americans, 605 of them nonsmokers, who were asked the following question, “Should the federal tax on cigarettes be raised by $1.25 to pay for health care reform?” Let p1 and p 2 equal the proportions of nonsmokers and smokers, respectively, who would say yes to this questions. Given that y1 = 351 nonsmokers and y 2 = 41 smokers said yes, a) With α = 0.05, test H0 : p1 = p 2 against H1 : p1 ≠ p 2 . State your conclusion in words. b) Find a 95% confidence interval for p1 − p 2 . State the meaning of this confidence interval in words. Is this in agreement with the conclusion of part a)?

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c) Find a 95% confidence interval for p, the proportion of adult Americans who would say yes. State the meaning of this confidence interval in words.

5. Bowl A contains 100 red balls and 200 white balls; bowl B contains 200 red balls and 100 white balls. Let p denote the probability of drawing a red ball from a bowl, but say p is unknown because it is unknown whether bowl A or bowl B is being used. We shall test the simple null hypothesis H0 : p = 1/3 against the simple alternative hypothesis H1 : p = 2/3. Draw three balls at random one at a time and with a replacement from the selected bowl. Let X equal the number of red balls drawn. Then let the rejection region be R = {x: x = 2, 3}. What are the values of α and β , the probabilities of Type I and Type II errors, respectively?

6. A new purification unit is installed in a chemical process. Before its installation, a random sample yield the following data about the percentage of impurity: x1 = 9.85,

S12 = 81.73, and n1 = 10. After the installation, a random sample resulted

in x 2 = 8.08,

S 22 = 78.46, and n 2 = 8.

a) Can you conclude that the two variances are equal? State your conclusion in words. Use α = 0.05. b) Assume equal variances, can you conclude that the new purification device has reduced the mean percentage of impurity? State your conclusion in words. Use

α = 0.05. 7. Let p equal the proportion of Chinese who favor the death penalty. If a random sample of n = 1234 Chinese yielded y = 864 who favored the death penalty, a) give a point estimate of p. b) Find an approximate 95% confidence interval for p. State the meaning of this confidence interval in words.

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8. The following data represent the running times films produced by 2 motion-picture companies: Company

Time (minutes)

1

102

86

98

109

92

2

81

165

97

134

92

87

114

Test the hypothesis that the average running time of films produced by company 2 exceeds the average running time of films produced by company 1 by 10 minutes, against the one-sided alternative that the difference is less than 10 minutes. State your conclusion in words.

Use a 0.1 level of significance and assume the

distributions of times to be approximately normal with unequal variances. 9. Let X 1 , X 2 , L, X n be a random sample of size n from a normal distribution. If cS is an unbiased estimator for σ , where

∑ (X n

S=

i =1

− X)

2

i

n −1

,

2 find an expression for c . (Hint: Recall the distribution of ( n − 1) S

σ

2

is χ n2−1 .)

10. Among the components used in producing a computer, 2 percent of the components are defective and the rest are good; 60 percent of the components are bought from the external sources and the company makes the rest. Of those bought from the external sources and the company makes the rest, 97 percent are good and the rest are defective. What is the probability that a randomly selected components is a) neither company- made nor good b) bought from the external source given that it is defective?

11. A computer center has three printers, A, B and C which print at different speeds. Programs are routed to the first available printer. The probability that a program is routed to printers A, B and C are 0.5, 0.3 and 0.1, respectively. Occasionally a printer will jam and destroy a printout. The probability that printers A, B and C will jam are

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0.01, 0.05, and 0.04, respectively. Your program is destroyed when a printer jams. What is the probability that printer A is involved? Printer B is involved? Printer C is involved?

12. If A, B and C are mutually independent, show the events A’ and (Bn C’) are independent. Also show that A’, B’ and C’ are mutually independent.

13. If a person visits his dentist, suppose that the probability that he will have his teeth cleaned is 0.44, the probability that he will have a cavity filled is 0.24, the probability that he will have a tooth extracted is 0.21, the probability that he will have his teeth cleaned and a cavity filled is 0.08, the probability that he will have his teeth cleaned and a tooth extracted is 0.11, the probability that he will have a cavity filled and a tooth extracted is 0.07, and the probability that he will have his teeth cleaned, a cavity filled, and a tooth extracted is 0.03. What is the probability that a person visiting his dentist will have at least one of these things done to him?

14. Given the values of the joint probability distribution of X and Y shown in the table. x

y

0

1

2

0

1/12

1/6

1/24

1

1/4

1/4

1/40

2

1/8

1/20

3

1/120

a) Find P( X > Y ) . b) Find the marginal density of X. c) Find the marginal density of Y. d) Find the conditional distribution of Y given X = 0 . e) Find the covariance of X and Y.

15. The joint probability density of X and Y is given by

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1  f ( x, y ) =  y  0

0< x < y<1

,

elsewhere

a) Verify that f ( x , y ) is a joint density function. b) Find the marginal density of X. c) Find the marginal density of Y. d) Find E ( X / Y ) .

16. The joint probability density of X and Y is given by 24 xy f ( x, y ) =   0

0 < x < 1, 0 < y < 1, x + y < 1 elsewhere

.

a) Verify that f ( x , y ) is a joint density function. 1 b) Find P( X + Y < ) . 2

17. The joint probability density of X and Y is given by

e − x− y f ( x, y ) =   0

x > 0, y > 0 elsewhere

,

a) Verify that f ( x , y ) is a joint density function. b) Find the covariance of X and Y.

18. If 7 cards are dealt from an ordinary deck of 52 playing cards, a) what is the probability that exactly 2 of them will be picture cards (J, Q, or K)? b) What is the probability if the 7 cards (2 picture cards and 5 non-picture cards) had been picked at random with replacement from the deck?

19. Flaws in a certain type of drapery material appear on the average of one in 150 square feet. What is the probability of at most one flaw in 225 square feet?

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20. The number of planes arriving per day at a small private airport is a random variable having a Poisson distribution with λ = 28 .8 . What is the probability that the time between two such arrivals is at least 1 hour?

21. A certain kind of appliance requires repairs on the average once every 2 years. What is the probability that such an appliance will work at least 3 years without requiring repairs? 22. The lifetime in weeks of a certain type of transistor is known to follow a gamma distribution with mean 10 weeks and standard deviation

50 weeks.

a) What is the probability that the transistor will last at most 50 weeks? b) What is the probability that the transistor will not survive the first 10 weeks? 23. In a photographic process, the developing time of prints may be looked upon as a random variable having the normal distribution with µ = 15.40 seconds and σ = 0.48 second. Find the probabilities that the time it takes to develop one of the prints will be a) at least 16.00 seconds; b) at most 14.20 seconds; c) anywhere from 15.00 to 15.80 seconds. 24. Let X be uniformly distributed and X can take values anywhere between 0 and 100. Find the mean and variance of X. 25. A random sample of size 64 is taken from a normal population with µ = 51.4 and

σ = 6.8 . What is the probability that the mean of the sample will a) exceed 52.9; b) fall between 50.5 and 52.3; c) be less than 50.6?

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26. A random sample of size 100 is taken from a normal population with σ = 25. What is the probability that the mean of the sample will differ from the mean of the population by 3 or mo re either way? 27. If X 1 , X 2 , L, X 16 is a random sample of size n = 16 from the normal distribution N (50,100) , find 16   a) P 796.2 ≤ ∑ ( X i − 50) 2 ≤ 2630 . i =1   16   b) P 726.1 ≤ ∑ ( X i − X ) 2 ≤ 2500 . i =1  

28. A process yields 10% defective items. If 100 it ems are randomly selected from the process, what is the probability that the number of defectives a) exceeds 13? b) is less than 8? 29. A study is to be made to estimate the proportion of residents in a certain city and its suburbs who favor the construction of a nuclear power plant. How large a sample is needed if one wishes to be at least 95% confident that the estimate is within 0.04 of the true proportion of residents in this city and its suburbs that favor the construction of the nuclear power plant?

30. If X 1 , X 2 and X 3 are independent and have the means 4, 9, and 3 and the variances 3, 7, and 5, find the mean and the variance of Y = 2 X 1 − 3 X 2 + 4 X 3 .

31. Let X 1 and X 2 be a random sample of size 2 from a distribution with probability density function f ( x ) = 6 x (1 − x ), 0 < x < 1 . Verify that f (x ) is a probability density function. Find the mean and the variance of Y = X 1 + X 2 .

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