MTH 110 Chapter 7 Practice Test Problems (FA06).tst

MTH 110 Chapter 7 Practice Test Problems

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1) The manager of a small retail store counted the number of sales each hour during a 60 -hour week. The frequency distribution is given below. Number of sales Number of during hour occurrences 6 25 7 20 8 10 9 0 10 5 The relative frequency of seven sales during an hour is 1 A) . 3 B)

7 . 40

C)

1 . 4

D)

7 . 60

E) none of the above 2) An urn contains four white balls and three red balls. The balls are drawn from the urn, one at a time without replacement, until two white balls have been drawn. Let X be the number of the draw on which the second white ball is drawn. The values that X may be are A) 2, 3, 4, 5. B) 1, 2, 3, 4, 5. C) 2, 3, 4, 5, 6, 7. D) 1, 2, 3, 4, 5, 6. E) none of the above Solve the problem. 3) Eighty families having four children each were surveyed and the number of girls was recorded for each. The table shows the resulting frequency distribution. number of number of girls occurrences 0 5 1 18 2 32 3 21 4 4 Determine the corresponding relative frequency distribution.

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4) An urn contains three white balls and two red balls. The balls are drawn from the urn, one at a time without replacement, until a white ball is drawn. Let X be the number of the draw on which a white ball is drawn for the first time. Determine the probability distribution for X. A pair of dice is tossed and the number of dots on the uppermost faces are observed. Let X denote the random variable defined as the larger of the two numbers . (If the two numbers are equal, then the value of X is either of them.) 5) Determine the probability distribution of X. 6) Compute Pr(X ≤ 3). Solve the problem. 7) A jury of 6 people is to be selected from an available pool of eight women and four men. Let X denote the number of men in the selected jury. Determine the probability distribution of X. 8) A coin is flipped three times; give the probability distribution of the number of heads. 9) The number of customers waiting to be served in a certain bank was counted every 15 minutes for five hours. The results are given in the frequency distribution below. Number of Number of customers occurrences 0 2 1 2 2 3 3 5 4 4 5 3 6 1 (a) Determine the corresponding relative frequency distribution. (b) Draw a histogram for the relative frequency distribution. 10) The number of accidents per week on a certain highway was recorded over a 40 -week period, and the results are summarized below. Number of Number of accidents occurrences 0 7 1 12 2 7 3 8 4 4 5 2 (a) Determine the relative frequency distribution for the number of accidents per week. (b) What is the highest number of accidents per week. (c) What is the number of accidents per week with the highest frequency?

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11) Suppose that the probability that a federal income tax return contains an arithmetic error is 0.2. If 10 federal income tax returns are selected at random, the probability that fewer than two of them contain an arithmetic error is A) 0.20. B) (0.8)10 + (0.2)10. C) 1 - [ (0.8)10 + 10(0.2)(0.8)9 ]. D) (0.8)10 + 10(0.2)(0.8)9 . E) none of the above 12) Let X denote the number of boys in a family with four children. Pr(X ≥ 3) is 5 . A) 16 B)

11 . 16

C)

2 . 3

D)

1 . 4

E) none of the above Solve the problem. 13) A single die is tossed five times. Find the probability that either a one or a two appears exactly four times. 14) A baseball player hits a home run with probability .02. What is the probability that he will hit a home run in a given game with four times at bat? Round to the nearest five decimal places. 15) A coin is flipped 10 times, and the number of heads is observed. What is the probability that heads appears nine times, given that it appears at least nine times? 16) The probablility for receiving an A in Calculus is given to be 20% in a class of 1000. If 20 students are chosen (with replacement) from the class, what is the probability that five of them received Aʹs? Round to the nearest four decimal places. A box contains nine balls labeled 1 through 9. A game consists of randomly drawing five balls from the box in succession, with replacement, and recording the number of each ball drawn. If two even numbers are recorded, the player wins the game. 17) What is the probability of winning this game? Round to the nearest four decimal places. A box contains 26 balls each of which is labeled with a different letter of the alphabet. An experiment consists of randomly drawing 10 of these balls and recording the letter of each ball drawn. 18) Compute the probability that two of the 10 letters are vowels. Round to the nearest four decimal places.

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Solve the problem. 19) A test for a certain drug produces a false positive 22% of the time. If a person who does not use the drug takes the test six times, what is the probability the person tests positive at least four times? Round to the nearest three decimal places. 20) The manager of a small retail store counted the number of sales each hour during a 60 -hour week. The frequency distribution is given below. Number of sales Number of during hour occurrences 6 25 7 20 8 10 9 0 10 5 The average number of sales during an hour is A) 6. 1 B) 8 . 2 C) 8. D) 7. E) none of the above 21) Consider the probability distribution below of the number of tails in 4 tosses of a fair coin. k Pr(X = k) 1 0 16 1

4 16

2

6 16

3

4 16

4

1 16

The mean is A) 1.75 B) 1.5. C) 1.0. D) 3.0. E) none of the above

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22) A luncheon at the cafeteria sold x adult tickets at $6 each and y childrenʹs tickets at $3 each. The average (arithmetic mean) revenue per ticket was 9xy A) x+y B)

6x+3y . x+y

C)

6x+3y . 9

D)

6x+3y x*y

E) none of the above 23) If three distinct positive integers have an average (arithmetic mean) of 80, and if the smallest of the three integers is 60, then the largest of the three integers can be at most A) 119. B) 100. C) 120. D) 80. E) none of the above Solve the problem. 24) Determine the expected value of the random variable X whose probability distribution is given below. k Pr(X = k) 0 0.5 1 0.2 2 0.3 3 0.1

25) Determine the expected value of the random variable X whose probability distribution is given below. k Pr(X = k) 0 0.2 1 0.4 2 0.3 3 0.1

26) Your grades on the four exams this term are 100, 75, 80, and 60. If the final test grade is worth 1.5 times a regular test, and the grade is a percentage, can you still get an average of 90? 27) Suppose a basketball team averages 82 points per game for 20 games and then averages 86 points per game for 10 games. If the team scores 96 and 88 points in the next two games, how many points per game did the team average over the 32 games?

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28) A shop supervisor counted the number of accidents each week during a year. The frequency distribution is given below. Number of accidents Number of during week occurrences 0 20 1 15 2 10 3 4 4 3 (a) Determine the probability distribution for the possible number of accidents during the week. (b) Find the expected number of accidents during the week. Round to the nearest three decimal places. 29) Consider the probability distribution below of the number of tails in 4 tosses of a fair coin. k Pr(X = k) 1 0 16 1

4 16

2

6 16

3

4 16

4

1 16

(a) Find the mean. (b) Find the expected number of tails during the experiment. 30) A multiple-choice test consists of five questions, each of which has a choice of four answers (only one of which is correct). A student guesses answers at random. Let X denote the number of questions the student answers correctly. (a) Find the probability distribution for X. (b) What is the expected value of X? 31) The variance of a probability distribution A) is large if the mean is large. B) is always smaller than the mean. C) is small if the mean is large. D) has little practical value. E) none of the above

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32) Consider the probability distribution below: k Pr(X = k) 0.2 -10 20 0.6 25 0.2

The mean is A) 20. B) 25. C) 15 D) 35 E) none of the above 33) Consider the probability distribution below: k Pr(X = k) 0.2 -10 20 0.6 25 0.2

The variance is A) 140 B) 200 C) 160 D) 35 E) none of the above

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34) Consider the probability distribution below: k Pr(X = k) 0.1 -2 0 0.2 1 0.1 2 0.2 3 0.4 The variance is A) 3.05. B) 0.0. C) 2.65. D) 1.18. E) none of the above 35) A certain probability distribution has mean 100 and variance 5. The standard deviation is A)

5.

B) 25. C) 20. D) 500. E) none of the above 36) Suppose that a probability distribution has mean 20 and standard deviation 3. The Chebychev inequality states that the probability that an outcome lies between 16 and 24 is 1 A) at most . 4 1 B) at least . 4 C) less than D) at least

7 . 16

7 . 16

E) none of the above 3 37) A binomial random variable with n = 1000 and p = has a variance of 4 A)

187.5.

B) 187.5. C)

750.

D) 750. E) none of the above

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Solve the problem. 38) Find the variance of k Pr(X = k) 0.2 -1 0 0.3 1 0.1 2 0.4

39) Find the mean, variance, and standard deviation for the probability distribution below. k Pr(X = k) 0 0.4 1 0.2 2 0.3 3 0.1

40) If X is a random variable such that E X2 =

15 5 and μ = , calculate σ2 . 2 2

41) Suppose that a probability distribution has mean 40 and standard deviation 2. Use the Chebychev inequality to estimate the probability that an outcome lies between 30 and 50. 42) Let X be the number of heads obtained when a fair coin is tossed 1,000 times. Using the Chebychev inequality, estimate the probability that X lies between 450 and 550. 43) Student A earned the following course grades during his freshman year: 4, 4, 4, 3, 3, 3, 3, 3, 2, 2. Student B earned the following course grades during his freshman year: 4, 4, 3, 3, 3, 2, 1, 1, 0, 0. (a) Compute the means and variances for each student. (b) Which student has the better grade point average? (c) Which student was more consistent? Explain. 44) Eighteen workers were sampled, and their days absent from work last year were recorded. The data are: 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 4, 4, 4, 4, 5, 5, 5, 6. (a) Determine the frequency distribution for these data. (b) Find the sample mean. (c) Find the sample variance.

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45) Which of the following statements is true? A) A normal curve is flatter when its standard deviation is small. B) A normal curve is completely described by its standard deviation. C) Some normal curves are not symmetric. D) There are many different normal curves with the same mean. E) none of the above 46) If Z is the standard normal random variable, then Pr( Z≤ 0.5) is A) 0.7723 B) .2277 C) 0.6915. D) 0.3085 E) none of the above 47) If Z is the standard normal random variable, then Pr( Z ≥ 0.6) is A) 0.7257. B) 0.5987. C) 0.2743. D) 0.2254. E) none of the above 48) Consider the normal distributions drawn below(with different scales). (a) (b)

They both have μ = 20 and the area of the shaded region of each is 0.90. Which of the following holds? A) There is not enough information. B) x 1 x 2 D) x 1 = x 2 E) x 1 ≥ x 2

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49) If X is the standard normal random variable, then Pr( -1.5 ≤ X ≤ 0) is A) 0.9332. B) 0.4332. C) 0.5000. D) 0.0668. E) none of the above 50) If Z is the standard normal random variable and Pr( -z ≤ Z ≤ z) = 0.6578, then z is A) 0.45. B) 0.40. C) 1.00. D) 0.95. E) none of the above 5 51) If X is a normal random variable with mean 12 and standard deviation , an x-value of 14 corresponds to a 4 standard value of A) 0.8. B) 2.5. C) 5.5. D) 1.6. E) none of the above 52) Suppose X is the random variable with a normal distribution with mean μ = 100 and the 95th percentile equal to 116.5.The value of σ is A) -10. B) 0.95. C) 16.5. D) 10. E) none of the above 53) The IQ of adults in a certain large population is normally distributed with mean 100 and standard deviation 10. If a person is chosen at random from this group, the probability that the personʹs IQ is less than 85 or greater than 110 is A) 0.2255. B) 0.7745. C) 0.9081. D) 0.0919. E) none of the above

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54) Tax returns are monitored by a state if the reported yearly income is in the top 15% of incomes in that state. Assume income is normally distributed with mean μ = $28,000 and standard deviation σ = $5000. The minimum income in the monitored group is A) $23,000. B) $22,750. C) $33,200. D) $33,000. E) none of the above 55) The lifetimes of a certain model of televisionʹs picture tubes are normally distributed with μ = 48 months and σ = 8 months. The manufacturer wants to issue a warranty that will be written so that about 92% of the picture tubes will outlast the warranty. For how many months should the picture tubes be guaranteed? A) 44.16 B) 59.20 C) 40.64 D) 36.80 E) none of the above Use a table to find the area of the shaded regions under the standard normal curve. 56)

57)

Find the value of z for which the area of the shaded region under the standard normal curve is given. 58)

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59)

If X has a normal distribution with μ = 100 and σ = 10, find the probability that X is 60) smaller than 96 or larger than 106. Solve the problem. 61) Let X be a random variable having a normal distribution with mean 25 and standard deviation 2. Find Pr(21 ≤ X ≤ 27). Find the value of x for which the area of the shaded region under the normal curve with μ = 100 and σ = 20 is given. 62)

63)

Solve the problem. 64) Find the 99th percentile of the standard normal distribution. 65) Suppose X is a random variable having a normal distribution with σ = 10 and 96th percentile of 22.5. Find μ. 66) The lifetimes of a certain model of washing machine are normally distributed with μ = 36 months and σ = 4 months. The manufacturer wants to issue a warranty that will be written so that about 96% of the machines will outlast the warranty. For how many months should the machines be guaranteed?

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67) In filling 10 oz. jars with instant coffee, the machine dispenses a normally distributed amount of coffee with μ = 10 oz. and σ = 0.1 oz. How many ounces of coffee should a jar hold so that no more than 5% of the jars overflow? The weights of men in a certain large group is normally distributed with μ = 160 lb. and σ = 15 lb. If X represents the weight of a man selected at random from the group, determine the following probabilities. 68) The man weighs more than 190 lb. 69) The man weighs less between 160 and 190 lb. 70) The man weighs less than 130 lb.

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Answer Key Testname: MTH 110 CHAPTER 7 PRACTICE TEST PROBLEMS (FA06)

1) A 2) A 3) number of relative girls frequency 0 0.0625 1 0.2250 2 0.4000 3 0.2625 4 0.0500 4) k Pr(X = k) 3 1 5 2

3 10

3

1 10

5) k 1

6)

Pr(X = k) 1 36

2

3 36

3

5 36

4

7 36

5

1 4

6

11 36

1 4

7) k 0

Pr(X = k) 1 33

1

8 33

2

5 11

3

8 33

4

1 33 15


8) Number of heads 0

Probability 1 8

1

3 8

2

3 8

3

1 8

9) (a) No. customers 0 1 2 3 4 5 6 (b)

10) (a) No. accidents 0 1 2 3 4 5 (b) 5 (c) 1 11) D 12) A 10 13) 243

Rel. freq. 0.10 0.10 0.15 0.25 0.20 0.15 0.05

Rel. freq 0.175 0.300 0.175 0.200 0.100 0.050

14) ≈ 0.07763 10 15) 11 16)

20 (.2)5 (.8)15 ≈ .1746 5 16


17) 5 · 4 2 · 5 3 ≈ 0.3387 9 9 2 18) 5 21 1260 2 8 = ≈ 0.3831 26 3289 10 19) ≈ 0.024 20) D 21) A 22) B 23) A 24) 1.1 25) 1.3 26) no 27) 83.875 points per game 28) (a) Number of accidents during a week 0

(b)

Probability 5 13

1

15 52

2

5 26

3

1 13

4

3 52

59 ≈ 1.135 52

29) (a) 2 (b) 2

17


30) (a) k 0

(b)

Pr(X = k) 243 1024

1

405 1024

2

135 512

3

45 512

4

15 1024

5

1 1024

5 = 1.25 4

31) E 32) C 33) C 34) C 35) A 36) D 37) B 38) 1.41 39) mean = 1.1, variance = 1.09, standard deviation = 1.09 5 40) 4 41) p ≥ 0.96 42) p ≥ 0.9 43) (a) Student A: μ = 3.1, σ2 = 0.49 Student B: μ = 2.1, σ2 = 2.09 (b) A (c) A; the variance of his score is lower. fi 44) (a) x i 4 1 2 5 3 1 4 4 5 3 6 1 (b) x = 3 46 (c) s2 = ≈ 2.71 17 45) D 46) C 47) C 48) B 49) B 18


50) D 51) D 52) D 53) A 54) C 55) D 56) 0.3085 57) 0.7745 58) z = 1.4 59) -z = -1.55 and z = 1.55 60) 0.6186 61) 0.8185 62) x = 113 63) x = 119 64) ≈ 2.35 65) 5 66) 29 months 67) 10.165 oz. 68) 0.0228 69) 0.4772 70) 0.0228

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