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PROBABILITY & STATISTICS FINAL EXAMINATION REVIEW

Students may use a graphing or scientific calculator for this exam. No students are permitted to share calculators. Students will be provided a formula sheet. Students are not allowed to share formula sheets. This exam will cover the following material and sections from the text Elementary Statistics: Picturing the World (6th Edition) (Larson and Farber)

Chapter 5: Normal Probability Distributions

5.1 Intro to the Normal Distribution

5.2 Normal Distribution: Finding Probabilities

5.3 Normal Distributions: Finding Values

5.4 Sampling Distributions and the Central Limit Theorem

5.5 Normal Approximation to the Binomial

Chapter 6: Confidence Intervals

6.1 Confidence Intervals for the Mean (σ known)

6.2 Confidence Intervals for the Mean (σ unknown)

6.3 Confidence Intervals for the Population Proportion

6.4 Confidence Intervals for the Standard Deviation and Variance

Chapter 7: Hypothesis Testing

7.1 Introduction to Hypothesis Testing

7.2 Hypothesis Testing for the Mean (σ known)

7.3 Hypothesis Testing for the Mean (σ unknown)

7.4 Hypothesis Testing for the Population Proportion

7.5 Hypothesis Testing for the Standard Deviation

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Applications of the Normal Distribution Draw and shade area under the normal curve for the following:

1. The SAT exam scores are normally distributed with a mean of 1000 and a standard deviation of 200.

a. What is the probability that a student, selected at random, will have a score higher than

1300?

b. Find the probability that a student will score between 900 and 1200.

c. Applicants to a prestigious school are required to have SAT scores at the 90th percentile or higher in order to be considered for admission. What is the minimum score that must be achieved on the SAT for admission to this school?

d. Applicants to a different school get automatically rejected if they only score in the first quartile. What is the minimum score needed to escape elimination from this school?

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2. The life span of a machine is normally distributed with a mean of 10.2 years and a standard deviation of 1.7 years. The manufacturer wants to replace the machine if it breaks before the warranty period is over. The manufacturer is only willing to replace 5% of the machines. What should they set for the warranty length for this machine?

3. Earthquakes that measure on the Richter scale in California are normally distributed with a mean of 6.2 and a standard deviation of 0.5.

a. Find the probability that a random earthquake has a magnitude of 6.0 or higher.

b. Find the probability that a random earthquake measures between 5.8 and 7.1.

c. Is a magnitude of 8.25 unusual?

d. What is the percentile rank of an earthquake that measures 6.8 on the scale?

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e. Determine the 40th percentile of the magnitude of earthquakes.

f. Determine the magnitude that would make up the middle 85% of the data.

4. The average mpg (miles per gallon) of a new model of motorcycle is known to be normally distributed with mean μ=27.4mpg and standard deviation σ=2.9 mpg.

a. Draw a normal curve with the parameters labeled. Shade the region that represents the proportion of mpgs between 23.7 and 29.3.

b. Suppose the area under the normal curve between 23.7 and 29.3 is 0.6419. Provide an interpretation of this result in context of the problem.

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5. Find the area under the standard normal curve to the left of z=1.25

a. 0.2318 b. 0.7682 c. 0.8944 d. 0.1056

6. Find the area of the standard normal curve between z = -1.5 and z = 2.5. a. 0.9270 b. 0.6312 c. 0.9831 d. 0.7182

7. Find the z-scores for which 90% of the distribution’s area lies between –z and z.

a. (-0.99,0.99) b. (-1.96, 1.96) c. (-2.33,2.33) d. (-1.645,1.645)

8. For a standard normal curve, find the z-score that separates the bottom 90% from the top

10%.

a. 1.52 b. 2.81 c. 0.28 d. 1.28

9. IQ test scores are normally distributed with a mean of 105 and a standard deviation of 15. An individual’s IQ score is found to be 126. Find the z-score corresponding to this value.

a. -1.4 b. 0.71 c. 1.4 d. -0.71

10. Determine the area under the standard normal curve that lies between z=0.7 and z=1.4.

a. 16.12% b. 91.92% c. 75.8% d. 24.2%

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11. A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 460 seconds and a standard deviation of 50 seconds. Find the probability that a randomly selected boy in secondary school can run the mile is less than 345 seconds.

a. 0.4893 b. 0.9893 c. 0.5107 d. 0.0107

12. The length of time it takes college students to find a parking spot in the library parking lot follows a normal distribution with a mean of 3.0 minutes and a standard deviation of 1 minute. Find the probability that a randomly selected college student will take between 1.5 and 4.0 minutes to find a parking spot in the library lot.

a. 0.7745 b. 0.2255 c. 0.0919 d. 0.4938

13. The tread life of a particular brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 1400 miles. What warranty should the company use if they want 96% of the tires to outlast the warranty?

a. 62,450 miles b. 61,400 miles c. 58,600 miles d. 57,550 miles

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Normal Approximation to the Binomial Distribution (5.5) As the number of trials increases, the binomial probability formula becomes more difficult to use. However, the binomial probability distribution becomes more and more symmetric. Therefore, we can use a normal curve to approximate these probabilities. General Rule: The distribution is considered approximately normal if np>5 and nq>5 where the mean is given by μx =np and the standard deviation is given by: σx= npq

Continuity correction:

Write the continuity correction for each of the following:

14. The probability of at most 40 15. The probability of exactly 12

16. The probability of at least 25 17. The probability of more than 17

18. The probability of a data value being at least 30 and at most 50.

19. The probability of fewer than 86 Use the following for questions 20-23. A study found that 70% of all US Households have cable tv. We collect a sample of 500 households and ask if they have cable tv.

20. How do we know that we are able to use the normal curve to approximate probabilities of this sample?

21. What is the probability that in this sample of 500, fewer than 340 have cable tv?

22. What is the probability that in this sample of 500, exactly 340 have cable tv?

23. What is the probability that in this sample of 500, more than 400 had cable tv?

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Sampling Distribution of the Sample Mean (5.4): The probability distribution of all possible values of the random variable x , computed from a sample size n from a population with mean μ and standard deviation σ.

24. A small apartment building has 3 different apartments. a. Use your calculator to find the mean and standard deviation.

b. Now, form all sample sizes where n=2 and find the means of each: c. Use your calculator to find the mean and standard deviation of these means. Thus, the mean of the sample means, denoted as µx = µ and the standard deviation of

the sample means, denoted as σ x =σn

If the population is normally distributed, then the sample means will be normally distributed. If the population is not normal, then the sample means will still be normally distributed if the sample size is 30 or more.

Apartment People Unit A 5 Unit B 1 Unit C 3

Samples Mean AA

AB

AC

BA

BB

BC

CA

CB

CC

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25. The heights of kindergarten children are normally distributed with a mean of 39 inches and a

standard deviation of 2 inches. If one child is selected at random, what is the probability that this child is taller than 41 inches?

26. Suppose that we have a class of 30 kindergarten children. What is the probability that the mean height of all the children in this class exceeds 41 inches?

27. The mean HSL cholesterol level for a healthy female in her 20’s is 53. If the HSL-cholesterol level is normally distributed with a standard deviation of 13.4, answer the following:

a. What is the probability that a randomly selected female in her twenties has a HSL-cholesterol level above 60?

b. What is the probability that among a sample of 15 females in their twenties, the mean cholesterol level is above 60?

c. What is the probability that among a sample of 20 females in their twenties, the mean HSL-cholesterol level is above 60?

28. The graph of a normal curve is given. Use the graph to identify the value of μ and σ.

μ =________________

σ = ________________

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29. A random variable x is normally distributed with μ=60. Convert the value of x to a z-score.

X=60, σ=6 a. 1 b. 6 c. 10 d. 0

30. True or False: The area under the normal curve drawn with regard to the population parameters is the same as the probability that a randomly selected individual of a population has these characteristics.

a. True b. False

31. Given a distribution that follows a standard normal curve, what does the graph of the curve do as z increases in the positive direction?

a. The graph of the curve eventually intersects the horizontal axis b. The graph of the curve approaches 1. c. The graph of the curve approaches 0. d. The graph of the curve approaches an inflection point.

32. The normal density curve is symmetric about:

a. The horizontal axis b. A point located one standard deviation from the mean c. An inflection point d. Its mean

33. Find the area under the standard normal curve to the right of z= -1.25

a. 0.7193 b. 0.6978 c. 0.5843 d. 0.8944

34. Use the standard normal distribution to find P(-2.50<z<1.50).

a. 0.5496 b. 0.6167 c. 0.8822 d. 0.9270

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35. Find the z-score that is less than the mean and for which 70% of the distribution’s area lies to

its right.

a. 0-0.98 b. -0.47 c. -0.81 d. -0.52

36. A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $400 per month and a standard deviation of $50 per month. Find the probability that in a randomly selected month, the amount spent on personal phone calls was below $250.

a. 0.9987 b. 0.3750 c. 0.6250 d. 0.0013

37. The amount of soda a dispensing machine pours into a 12 oz can of soda follows a normal distribution with a mean of 12.48 oz and a standard deviation of 0.32 oz. The cans only hold 12.8 oz. of soda. Every can that has more than 12.8 oz of soda poured into it causes a spill and the can needs to go through a special cleaning process before it can be sold. What is the probability that a randomly selected can will need to go through the cleaning process?

a. 0.8413 b. 0.3413 c. 0.1587 d. 0.6587

38. Suppose that prices of a certain model of new homes are normally distributed with a mean of $150,000. Find the percentage of buyers who paid between $150,000 and $154,800 if the standard deviation is $2400. a. 47.7% b. 99.7% c. 68% d. 34%

39. Assuming that all conditions are met to approximate a binomial probability distribution with the standard normal distribution, then to compute P(12<x<15) from the binomial distribution we must compute _________________ as the normal approximation.

a. P(12.5<x<14.5) b. P(x>12.5) and P(x<14.5) c. P(11.5<x<15.5) d. P(x>15.5) and P(x<11.5)

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40. Assuming that all conditions are met to approximate a binomial probability distribution with

the standard normal distribution, then to compute P(x>19) from the binomial distribution we must compute _________________ as the normal approximation.

a. P(x<18.5) b. P(x>19.5) c. P(x<18.9) d. P(x>18.5)

41. If the probability of a newborn child being female is 0.5, find the probability that in 100 births, 55 or more will be female. Use the normal distribution to approximate the binomial distribution.

a. 0.1841 b. 0.0606 c. 0.8159 d. 0.7967

42. A telemarketer found that there was a 1% chance of a sale from his phone solicitations. Find the probability of getting a 5 or more sales for 1000 telephone calls.

a. 0.0401 b. 0.0871 c. 0.9599 d. 0.8810

43. The amount of money collected by the snack bar at a large university has been recorded daily for the past 5 years. Records indicate that the mean daily amount collected is $2750 and the standard deviation is $350. The distribution is skewed to the right due to several high volume days. Suppose that 100 days were randomly selected and the average amount collected from those days was recorded. Which of the following describes the sampling distribution of the sample mean?

a. normally distributed with a mean of $275 and a standard deviation of $35 b. normally distributed with a mean of $2750 and a standard deviation of $35 c. skewed to the right with a mean of $2750 and a standard deviation of $350 d. normally distributed with a mean of $2750 and a standard deviation of $350

44. The mount of corn chips dispensed into a 12-oz bag by the dispensing machine has been identified at possessing a normal distribution with a mean of 12.5 oz and a standard deviation of 0.3 oz. Suppose 400 bags of chips were randomly selected from this dispensing machine. Find the probability that the sample mean weight of these 400 bags exceeded 12.6 oz.

a. approximately 0 b. 0.1915 c. 0.6915 d. 0.3085

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45. The lengths of pregnancies are normally distributed with a mean of 263 days anad a standard

deviation of 25 days. If 100 women are randomly selected, find the probability that they have a mean pregnancy between 263 days and 265 days.

a. 0.2881 b. 0.7881 c. 0.5517 d. 0.2119

47. A farmer was interested in determining how many grasshoppers were in his field. He knows that they distribution of grasshoppers may not be normally distributed in his field due to growing conditions. As he drives his tractor down each row he counts how many grasshoppers he sees flying away. After 40 rows he figures the mean number of flights to be 57 with a standard deviation of 12 flights. What is the probability that the farmer will count 52 or fewer as the flights average in the next 40 rows down which he drives his tractor?

a. 0.0041 b. 0.9959 c. 0.0410 d. 0.4959

48. Professor WhataGuy took a survey of his Introduction to Statistics class of 420 students. One of the questions was “Will you take another mathematics class?” The results showed that 252 of the students said yes. What is the sample proportion p

! ?

a. 0.6 b. 0.42 c. 0.775 d. 0.252

49. Smith is a weld inspector at a shipyard. He knows from keeping track of good and substandard welds that for the after shift 5% of all welds done will be substandard. If Smith checks 300 of the 7500 welds completed that shift, would it be unusual for Smith to find 30 substandard welds?

a. Yes b. No

50. The board of examiners that administers the real estate broker’s examination in a certain state found that the mean score on the test of 523 and the standard deviation was 72. If the board wants to set the passing score so that only the best 80% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

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51. A local motel has 200 rooms. The occupancy rate for the winter months is 60%. Find the

probability that in a given winter month, at least 140 rooms will be rented. Use the normal distribution to approximate the binomial distribution.

Constructing a Confidence Interval

52. A local Sears retail store keeps records of the value (in dollars) of items stolen and recovered from shoplifters.

144.95 144 542.88 346.95 149.99 402 34 119.96 134.98 16 100.97 37.5 209 38

55 23.99 17.99 145.94 99.97 449.99 17.47 24.99 39.99 37.99 529.88 9.99 274.89 379.99

162.97 24.99 229.99 16.96 101.91 389.96 79.99 330.96

a. Compute a point estimate for the population mean value in dollars of all items stolen and recovered from Sears. b. Construct a 95% confidence interval for the mean value of all items stolen and recovered from Sears.

c. Construct a 99% confidence interval of the mean value of all items stolen and recovered from Sears.

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53. A random sample of 50 people found that the mean travel time to work is 24.2 minutes. Assuming that the population standard deviation is 18.5 minutes, construct a 95% confidence interval. Interpret this interval.

54. Determining a sample size: How many subjects are needed to estimate the time American teens spend on the internet each week within 0.5 hours with 95% confidence? Studies indicate that σ=6.6 hours.

Hypothesis Testing

55. According to a detailed study, 73.8% of females aged 18-29 years exercise regularly. You think this percentage should be higher, given women today. So, you sample 1000 women.

a. What conclusion could be made if you find that 750 exercise regularly?

b. What conclusion could be made if you find that 920 are exercising regularly?

56. What is the definition of hypothesis testing?

57. What is the definition of the null hypothesis?

58. What is the definition of the alternate hypothesis?

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For problems 59 – 61, write the null and alternate hypotheses.

59. A poll conducted in 1995 showed that 74% of Americans felt that men were more aggressive than women. A researcher claims that the percentage is different today.

60. A light bulb package states that the average life of the light bulb will be 500 hours. A customer advocate groups wants to know if the mean life of the light bulb is less than 500 hours.

61. A new antibiotic says that 2% of children taking the antibiotic have had some side effects. A researcher from the FDA wants to test the claim that the percentage is actually more than only 2%.

62. To test H0 :µ = 40H1 :µ < 40 , a random sample of 25 is obtained from a population that is known to

be normally distributed with σ=6.

a. The sample mean is determined to be x =42.3, compute the test statistic.

b. The researcher decides to test this hypothesis at the α=0.10 level of significance, determine the critical values.

c. Draw the normal curve that depicts this critical region.

d. Will the researcher reject the null hypothesis?

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63.The manufacturer of a particular type of ATM machine reports that the mean ATM withdrawal is $60. The manager of a convenience store with an ATM machine thinks that the mean withdrawal from his machine is less than this amount. He obtains a simple random sample of 35 withdrawals over the past year and finds the sample mean to be $52. Assume the population standard deviation is $13. Test the manager’s claim.

64. The area under a standard normal density curve with mean of 0 and a standard deviation of 1 is:

a. μ+3σ b. μ+2(3σ) c. 1 d. infinite

65. Find the area under the standard normal curve to the left of z=1.5

a. 0.9332 b. 0.5199 c. 0.0668 d. 0.7612

66. Find the area under the standard normal curve between z=1.5 and z=2.5.

a. 0.9938 b. 0.9816 c. 0.9332 d. 0.0606

67. The tread life of a certain brand of tire is a random variable best described by a normal distribution with a mean of 60,000 miles and a standard deviation of 1200 miles. What is the probability a particular tire of this brand will last longer than 58,800 miles?

a. 0.2266 b. 0.1587 c. 0.7266 d. 0.8413

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68. The length of time it takes college students to find a parking spot in the library parking lot

follows a normal distribution with a mean of 5.5 minutes and a standard deviation of 1 minute. Find the cutoff time which 75.8% of the college students exceed when trying to find a parking spot in the library parking lot.

a. 6.2 b. 6.3 c. 6.0 d. 5.8

69. Assuming that all conditions are met to approximate a binomial probability distribution with the standard normal distribution, then to compute the probability that at least 40 households have a gas stove from the binomial distribution we must compute _______________as the normal approximation:

a. P(x>40) b. P(x<40.5) c. P(x<40) d. P(x>39.5)

For the following, show all work for credit!

70. A survey was conducted to measure the heights of American males. In the 20-29 age group, heights are normally distributed with a mean of 69.2 inches and a standard deviation of 2.9 inches. What is the probability that a randomly selected male in this age group will be taller than 72 inches?

71. The lifespan of a machine is normally distributed with a mean of 12 years and a standard

deviation of 1.75 years. The manufacturer will replace a machine if it breaks before the guarantee period is over. If the manufacturer is willing to replace only 3% of the machines, find the time limit that the manufacturer should set for the guarantee period.

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72. A college requires that applicants have an ACT score in the top 13% of all test scores. The ACT tests scores are normally distributed with a mean of 21 and a standard deviation of 4.7. Find the lowest test score that a student could get and still meet the requirement of the college.

73. In clinical trials and advanced studies of a medication whose purpose is to reduce the pain associated with migraine headaches, only 2% of the patients in the study experienced weight gain as a side effect. Suppose a random sample of 600 users of this medication is obtained.

a. Explain why we can use the normal distribution to approximate the binomial

distribution for this problem. b. Use the normal distribution to approximate the probability that 20 or fewer will

experience weight gain.

74. Credit card balances are normally distributed with a mean of $4500 and a standard deviation of $1200. You randomly select 25 credit card balances. What is the probability that the mean of the sample of the 25 credit card balances is more than $5000?

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75. The mean rent of an apartment in a city is normally distributed with a mean of $1,300 and a standard deviation of $190. You randomly select 9 apartments. What is the probability that the mean of these 9 apartments is less than $1,175?

76. Suppose a simple random sample of n=100 households is obtained from a town with 5,000

households. It is known that 30% of the households plant a garden in the spring season.

a. Determine the mean and the standard deviation of the sample proportion, p! .

µp!=

σp!=

b. What is the probability that more than 37 households in the sample plant a garden in

the spring?

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77. The board of examiners that administers the real estate broker’s examination in a certain state found that the mean score on the test was 494 and the standard deviation was 72. If the board wants to set the passing score so that only the best 10% of all applicants pass, what is the passing score? Assume that the scores are normally distributed.

78. A recent survey found that 79% of all adults over 50 wear glasses for driving. You randomly

select 37 adults over 50, and ask if he or she wears glasses. Decide whether you can use the normal distribution to approximate the binomial distribution. If so, find the mean and standard deviation. If not, explain why.

79. The average score of all golfers for a particular course has a mean of 66 and a standard

deviation of 5. Suppose 100 golfers played the course today. Find the probability that the average score of the 100 golfers exceeded 67.

80. In a sample of 10 randomly selected women, it was found that their mean height was 63.4

inches. From previous studies, it is assumed that the standard deviation, σ, is 2.4. Construct the 95% confidence interval for the population mean.

a. (58.1,67.3) b. (59.7,66.5) c. (61.9,64.9) d. (60.8,65.4)

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81. Given z is the standard normal score, find each:

a. P(0 < z < 1.25) b. P(z > 1.45) c. P(z < 1.82) d. P(-1.34 < z < 2.50) e. P(1.45 < z < 2.4)

82. Find z so that

a. 5% is more than z b. 10% is less than z c. 94% is between –z and z

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83. Express Courier has found that the delivery time for packages is normally distributed with a mean of 14 hours and a standard deviation of 2 hours.

a. Find the probability that the package is delivered in less than 12 hours. b. Find the probability that the package is delivered between 13 and 16 hours.

c. Find the probability that the package is delivered between 15 and 17 hours. d. Find the value of the 90th Percentile. e. If a group of 16 packages is randomly selected, find the probability that the mean

delivery time is less than 12 hours.

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84. In Roosevelt Forest, rangers took random samples of aspen trees and measured the circumference at the base. Assume the measures are normally distributed and the population standard deviation is 4.62.

a. For 30 trees, the mean was 15.71. Find a 95% confidence interval for the population

mean circumference of all trees in Roosevelt Forest. b. For 90 trees, the mean was 15.58. Find a 99% confidence interval for the population

mean circumference of all trees in Roosevelt Forest. c. How many trees must be measured to obtain the mean within 0.2 inches at 95%

confidence? d. Repeat c for an error of 0.1 inches.

85. Suppose the rangers thought that the trees were larger than normal and had read

somewhere that the mean of all aspen trees was 15.25.

a. Write the alternative hypothesis to test the claim that the mean is greater than 15.25. b. Determine the critical value using α=0.05. c. Calculate the test statistic for the 30 trees with a mean of 15.71. d. Do we reject or fail to reject the null hypothesis?