Deck 7: Normal Curves and Sampling Distributions

Question 1

Statistical Literacy Does a raw score less than the mean correspond to a positive or negative standard score What about a raw score greater than the mean

Answer 1

If you have a raw score that is below the mean, then the corresponding z score (standard score) will be negative.
A raw score that is above the mean will have a standard score (z score) that is positive.
This is because the mean µ of the z distribution is 0, and so any score above it will be positive and any score below it will be negative.

Question 2

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
To the left of z = 0.72

Answer 2

First, we shade the area to be found on the standard normal distribution as shown in the figure:
(Area to the left of 0.72) = 0.7642

Question 3

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 2.15)

Answer 3

refers to the probability that z values lie to the left of - 2.15. Therefore,
.
After, finding the area, we shade the corresponding area under the standard normal curve.

Question 4

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P (0.73 z 3.12)

Question 5

Basic Computation: Find z Values In Problems 15-24, find the z value described and sketch the area described.
Find z such that 5.2% of the standard normal curve lies to the left of z.

Question 6

Estimating the Standard Deviation: Refrigerator Replacement Consumer Reports indicated that the average life of a refrigerator before replacement in = 14 years with a (95% of data) range from 9 to 19 years. Let x = age at which a refrigerator is replaced. Assume that x has a distribution that is approximately normal.
(a) Find a good approximation for the standard deviation of x vales. Hint: See Problem 31.
(b) What is the probability that someone will keep a refrigerator fewer than 11 years before replacement
(c) What is the probability that someone will keep a refrigerator more than 18 years before replacement
(d) Inverse Normal Distribution Assume that the average life of a refrigerator is 14 years with the standard deviation given in part (a) before it breaks. Suppose that a company guarantees refrigerators and will replace a refrigerator that breaks while under guarantee with a new one. However, that company does not want to replace more than 5% of the refrigerators under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)

Question 7

Statistical Literacy Describe how the variability of the
distribution changes as the sample size increases.

Question 8

Finance: European Growth Fund A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has more than 100 stocks. Let x be a random variable that represents the monthly percentage return for this fund. Based on information from Morningstar (see Problem 19), x has mean = 1.4% and standard deviation = 0.8%.
(a) Let's consider the monthly return of the stocks in the European growth fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume that x (the average monthly return on the 100 stocks in the European growth fund) has a distribution that is approximately normal Explain. Hint: See Problem 19, part (a).
(b) After 9 months, what is the probability that the average monthly percentage return
will be between 1% and 2% Hint: See Theorem 7.1 and the results of part (a).
(c) After 18 months, what is the probability that the average monthly percentage return
will be between 1% and 2%
(d) Compare your answers to parts (b) and (c). Did the probability increase as n (number of months) increased Why would this happen
(e) Interpretation: If after 18 months the average monthly percentage return
is more than 2%, would that tend to shake your confidence in the statement that = 1.4% If this happened, do you think the European stock market might be heating up Explain.

Question 9

Insurance: Claims Do you try to pad an insurance claim to cover your deductible About 40% of all U.S. adults will try to pad their insurance claims! (Source: Are You Normal , by Bernice Kanner, St. Martin's Press.) Suppose that you are the director of an insurance adjustment office. Your office has just received 128 insurance claims to be processed in the next few days. What is the probability that
(a) half or more of the claims have been padded
(b) fewer than 45 of the claims have been padded
(c) from 40 to 64 of the claims have been padded
(d) more than 80 of the claims have not been padded

Question 10

General: Approximations We have studied the normal approximation to the binomial distribution. Write a brief but complete essay in which you discuss and summarize the condition under which the approximation would be used, the formulas involved, and the assumptions made for the approximation. Give details and examples in your essay. How could you apply these statistical methods in your everyday life

Question 11

Statistical Literacy What is the value of the standard score for the mean of a distribution

Question 12

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
To the right of z = 1.52

Question 13

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 1.20)

Question 14

Statistical Literacy Consider a normal distribution with mean 30 and standard deviation 2. What is the probability a value selected at random form this distribution is greater than 30

Question 15

Basic Computation: Find z Values In Problems 15-24, find the z value described and sketch the area described.
Find z such that 55% of the standard normal curve lies to the left of z.

Question 16

Estimating the Standard Deviation: Veterinary Science The resting heart rate for an adult horse should average about = 46 beats per minute with a (95% of data) range from 22 to 70 beats per minute, based on the information from the Merck Veterinary Manual (a classic reference used in most veterinary colleges). Let x be a random variable that represents the resting heart rate for an adult horse. Assume that x has a distribution that is approximately normal.
(a) Estimate the standard deviation of the x distribution. Hint: See Problem 31.
(b) What is the probability that the heart rate is fewer than 25 beats per minute
(c) What is the probability that the heart rate is greater than 60 beats per minute
(d) What is the probability that the heart rate is between 25 and 60 beats per minute
(e) Inverse Normal Distribution A horse whose resting heart rate is in the upper 10% of the probability distribution of heart rates may have a secondary infection or illness that needs to be treated. What is the heart rate corresponding to the upper 10% cutoff point of the probability distribution

Question 17

Basic Computation: Central Limit Theorem Suppose x has a distribution with a mean of 8 and a standard deviation of 16. Random samples of size n = 64 are drawn.
(a) Describe the
distribution and compute the mean and standard deviation of the distribution.
(b) Find the z value corresponding to
.
(c) Find
.
(d) Interpretation Would it be unusual for a random sample of size 64 from the x distribution to have a sample mean greater than 9 Explain.

Question 18

Statistical Literacy Describe a normal probability distribution.

Question 19

Critical Thinking Let x be a random variable representing the amount of sleep each adult in New York City got last night. Consider a sampling distribution of sample means
.
(a) As the sample size becomes increasingly large, what distribution does the
distribution approach
(b) As the sample size becomes increasingly large, what value will the mean
of the
distribution approach
(c) What value will the standard deviation
of the sampling distribution approach
(d) How do the two
distributions for sample size n = 50 and n = 100 compare

Question 20

Recycling: Aluminum Cans One environmental group did a study of recycling habits in a California community. It found that 70% of the aluminum cans sold in the area were recycled.
(a) If 400 cans are sold today, what is the probability that 300 or more will be recycled
(b) Of the 4 00 cans sold, what is the probability that between 260 and 300 will be recycled

Question 21

Statistical Literacy What are the values of the mean and standard deviation of a standard normal distribution

Question 22

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
To the right of z = 0.15

Question 23

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 3.20)

Question 24

Statistical Literacy Suppose 5% of the area under the standard normal curve lies to the right of z Is z positive or negative

Question 25

Basic Computation: Find z Values In Problems 15-24, find the z value described and sketch the area described.
Find z such that 97.5% of the standard normal curve lies to the left of z.

Question 26

Estimating the Standard Deviation: Veterinary Science How much should a healthy kitten weigh A healthy 10-week-old (domestic) kitten should weigh an average of = 24.5 ounces with a (95% of data) range from 14 to 35 ounces. (See reference in Problem 33.) Let x be a random variable that represents the weight (in ounces) of a healthy 10-week-old kitten. Assume that x has a distribution that is approximately normal.
(a) Estimate the standard deviation of the x distribution. Hint: See Problem 31.
(b) What is the probability that a healthy 10-week-old kitten will weigh less than 14 ounces
(c) What is the probability that a healthy 10-week-old kitten will weigh more than 33 ounces
(d) What is the probability that a healthy 10-week-old kitten will weigh between 14 and 33 ounces
(e) Inverse Normal Distribution A kitten whose weight is in the bottom 10% of the probability distribution of weights is called undernourished. What is the cutoff point for the weight of an undernourished kitten

Question 27

Basic Computation: Central Limit Theorem Suppose x has a distribution with a mean of 20 and a standard deviation of 3. Random samples of size n = 36 are drawn.
(a) Describe the
distribution and compute the mean and standard deviation of the distribution.
(b) Find the z value corresponding to
.
(c) Find
.
(d) Interpretation Would it be unusual for a random sample of size 36 from the x distribution to have a sample mean less than 19 Explain.

Question 28

Statistical Literacy Binomial probability distributions depend on the number of trials n of a binomial experiment and the probability of success p on each trail. Under what conditions is it appropriate to use a normal approximation to the binomial

Question 29

Longevity: 90th Birthday It is estimated that 3.5% of the general population will live past their 90th birthday (Statistical Abstract of the United States, 112th Edition). In a graduating class of 753 high school seniors, what is the probability that
(a) 15 or more will live beyond their 90th birthday
(b) 30 or more will live beyond their 90th birthday
(c) between 25 and 35 will live beyond their 90th birthday
(d) more than 40 will live beyond their 90th birthday

Question 30

Statistical Literacy Under what conditions is it appropriate to use a normal distribution to approximate the
distribution

Question 31

Statistical Literacy Which, if any, of the curves in Figure 7-7 look(s) like a normal curve If a curve is not a normal curve, tell why.

Question 32

Basic Computation: z Score and Raw Score A normal distribution has = 30 and = 5.
(a) Find the z score corresponding to x = 25.
(b) Find the z score corresponding to x = 42.
(c) Find the raw score corresponding to z = 2.
(d) Find the raw score corresponding to z = 1.3.

Question 33

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
To the right of z = 1.22

Question 34

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 1.35)

Question 35

Statistical Literacy Suppose 5% of the area under the standard normal curve lies to the left of z Is z positive or negative

Question 36

Basic Computation: Find z Values In Problems 15-24, find the z value described and sketch the area described.
Find z such that 8% of the standard normal curve lies to the right of z.

Question 37

Insurance: Satellites A relay microchip in a telecommunications satellite has a life expectancy that follows a normal distribution with a mean of 90 months and a standard deviation of 3.7 monts. When this computer-relay microchip malfunctions, the entire satellite is useless. A large London insurance company is going to insure the satellite for $50 million. Assume that the only part of the satellite in question is the microchip. All other components will work indefinitely.
(a) Inverse Normal Distribution For how many months should the satellite be insured to be 99% confident that it will last beyond the insurance date
(b) If the satellite is insured for 84 months, what is the probability that it will malfunction before the insurance coverage ends
(c) If the satellite is insured for 84 months, what is the expected loss to the insurance company
(d) If the insurance company charges $3 million for 84 months of insurance, how much profit does the company expect to make

Question 38

Statistical Literacy
(a) If we have a distribution of x values that is more or less mound-shaped and somewhat symmetrical, what is the sample size needed to claim that the distribution of sample means
from random samples of that size is approximately normal
(b) If the original distribution of x values is known to be normal, do we need to make any restriction about sample size in order to claim that the distribution of sample means
taken from random samples of a given size is normal

Question 39

Statistical Literacy According to the empirical rule, approximately what percentage of the area under a normal distribution lies within I standard deviation of the mean within 2 standard deviations within 3 standard deviations

Question 40

Critical Thinking If x has a normal distribution with mean = 15 and standard deviation = 3, describe the distribution of
values for sample size n , where n = 4, n = 16 and n = 100. How do the
distributions compare for the various sample sizes

Question 41

Guarantee: Disc Players Future Electronics makes compact disc players. Its research department found that the life of the laser beam device is normally distributed, with mean 5000 hours and standard deviation 450 hours.
(a) Find the probability that the laser beam device will wear out in 5000 hours or less.
(b) Inverse Normal Distribution Future Electronics wants to place a guarantee on the players so that no more than 5% fail during the guarantee period. Because the laser pickup is the part most likely to wear out first, the guarantee period will be based on the life of the laser beam device. How many playing hours should the guarantee cover (Round to the next playing hour.)

Question 42

Statistical Literacy Look at the normal curve in Figure 7-8, and find , + , and .

Question 43

Basic Computation: z Score and Raw Score A normal distribution has = 10 and = 2.
(a) Find the z score corresponding to x = 12.
(b) Find the z score corresponding to x = 4.
(c) Find the raw score corresponding to z = 1.5.
(d) Find the raw score corresponding to z = 1.2.

Question 44

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
To the right of z = 2.17

Question 45

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 2.17)

Question 46

Critical Thinking: Normality Consider the following data. The summary statistics, histogram, and normal quantile plot were generated by Minitab.


(a) Does the histogram indicate normality for the data distribution Explain.
(b) Does the normal quantile plot indicate normality for the data distribution Explain.
(c) Compute the interquartile range and check for outliers.
(d) Compute Person's index. Does the index value indicate skewness
(e) Using parts (a) through (d), would you say the data are from a normal distribution

Question 47

Basic Computation: Find z Values In Problems 15-24, find the z value described and sketch the area described.
Find z such that 5% of the standard normal curve lies to the right of z.

Question 48

Statistical Literacy What is a population Give three examples.

Question 49

Critical Thinking Suppose x has a distribution with = 72 and = 8.
(a) If random samples of size n = 16 are selected, can we say anything about the
distribution of sample means
(b) If the original x distribution is normal , can we say anything about the
distribution of random samples of size 16 Find
.

Question 50

Statistical Literacy When we use a normal distribution to approximate a binomial distribution, why do we make a continuity correction

Question 51

Fishing: Billfish Ocean fishing for billfish is very popular in the Cozumel region of Mexico. In World Record Game Fishes (published by the International Game Fish Association), it was stated that in the Cozumel region, about 44% of strikes (while trolling) result in a catch. Suppose that on a given day a fleet of fishing boats got a total of 24 strikes. What is the probability that the number of fish caught was
(a) 12 or fewer
(b) 5 or more
(c) between 5 and 12

Question 52

Statistical Literacy What is the formula for the standard error of the normal approximation to the
distribution What is the mean of the
distribution

Question 53

Critical Thinking Look at the two normal curves in Figure 7-9 and 7-10. Which has the larger standard deviation What is the mean of the curve in Figure 7-9 What is the mean of the curve in Figure 7-10

Question 54

Critical Thinking Consider the following scores:
(i) Score of 40 from a distribution with mean 50 and standard deviation 10
(ii) Score of 45 from a distribution with mean 50 and standard deviation 5
How do the two scores compare relative to their respective distributions

Question 55

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
Between z = 0 and z = 3.18

Question 56

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 1.20)

Question 57

Basic Computation: Find Probabilities In Problems 5-14, assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities.
P (3 x 6); = 4; = 2

Question 58

Basic Computation: Find z Values In Problems 15-24, find the z value described and sketch the area described.
Find z such that 82% of the standard normal curve lies to the right of z.

Question 59

Statistical Literacy What is a random sample from a population Hint: See Section 1.2.

Question 60

Critical Thinking Consider two
distributions corresponding to the same x distribution. The first
distribution is based on samples of size n = 100 and the second is based on samples of size n = 225. Which
distribution has the smaller standard error Explain.

Question 61

Statistical Literacy Random sample of size 9 are taken from a normal distribution with mean 10 and standard deviation 3. Describe the shape of the resulting
sampling distribution. What are the mean and standard deviation of the sampling distribution

Question 62

Basic Computation: Probability Given that x is a normal variable with mean = 47 and standard deviation = 6.2, find
(a) p(x 60 )
(b) p ( x 50)
(c) p (50 x 60)

Question 63

Guarantee: Package Delivery Express Courier Service has found that the delivery time for packages is normally distributed, with mean 14 hours and standard deviation 2 hours.
(a) For a package selected at random, what is the probability that it will be delivered in 18 hours or less
(b) Inverse Normal Distribution What should be the guaranteed delivery time on all packages in order to be 95% sure that the package will be delivered before this time Hint: Note that 5% of the packages will be delivered at a time beyond the guaranteed time period.

Question 64

Critical Thinking Sketch a normal curve
(a) with mean 15 and standard deviation 2.
(b) with mean 15 and standard deviation 3.
(c) with mean 12 and standard deviation 2.
(d) with mean 12 and standard deviation 3.
(e) Consider two normal curves. If the first one has a larger mean than the second one, must it have a larger standard deviation as well Explain your answer.

Question 65

Critical Thinking Raul received a score of 80 on a history test for which the class mean was 70 with standard deviation 10. He received a score of 75 on a biology test for which the class mean was 70 with standard deviation 2.5. On which test did he do better relative to the rest of the class

Question 66

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
Between z = 0 and z = 1.93

Question 67

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P ( z 1.50)

Question 68

Basic Computation: Find Probabilities In Problems 5-14, assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities.
P (10 x 26); = 15; = 4

Question 69

Find z such that 95% of the standard normal curve lies to the right of z.

Question 70

Statistical Literacy What is a population parameter Give three examples.

Question 71

Critical Thinking Consider an x distribution with standard deviation = 12.
(a) If specifications for a research project require the standard error of the corresponding
distribution to be 2, how large does the sample size need to be
(b) If specifications for a research project require the standard error of the corresponding
distribution to be 1, how large does the sample size need to be

Question 72

Basic Computation: Normal Approximation to a Binomial Distribution Suppose we have a binomial experiment with n = 40 trials and a probability of success p = 0.50.
(a) Is it appropriate to use a normal approximation to this binomial distribution Why
(b) Compute and of the approximating normal distribution.
(c) Use a continuity correction factor to convert the statement r 23 successes to a statement about the corresponding normal variable x.
(d) Estimate P ( r 23).
(e) Interpretation Is it unusual for a binomial experiment with 40 trials and probability of success 0.50 to have 23 or more successes Explain.

Question 73

Grocery Stores: New Products The Denver Post stated that 80% of all new products introduced in grocery stores fail (are taken off the market) within 2 years. If a grocery store chain introduces 66 new products, what is the probability that within 2 years
(a) 47 or more fail
(b) 58 or fewer fail
(c) 15 or more succeed
(d) fewer than 10 succeed

Question 74

Statistical Literacy Is
an unbiased estimator for p when np 5 and np 5 Recall that a statistic is an unbiased estimator of the corresponding parameter if the mean of the sampling distribution equals the parameter in question.

Question 75

Basic Computation: Empirical Rule What percentage of the area under the normal curve lies
(a) to the left of
(b) between - and +
(c) between - 3 and + 3

Question 76

z Scores: First Aid Course The college physical education department offered an advanced first aid course last semester. The scores on the comprehensive final exam were normally distributed, and the z scores for some of the students are shown below:

(a) Which of these students scored above the mean
(b) Which of these students scored on the mean
(c) Which of these students scored below the mean
(d) If the mean score was = 150 with standard deviation = 20, what was the final exam score for each student

Question 77

Basic Computation: Finding Areas Under the Standard Normal Curve In Problems 13-30, sketch the areas under the standard normal curve over the indicated intervals and find the specified areas.
Between z = 2.18 and z = 1.34

Question 78

Basic Computation: Finding Probabilities In Problems 31-50, let z be a random variable with a standard normal distribution, Find the indicated probability, and shade the corresponding area under the standard normal curve.
P (1.20 z 2.64)

Question 79

Basic Computation: Find Probabilities In Problems 5-14, assume that x has a normal distribution with the specified mean and standard deviation. Find the indicated probabilities.
P (50 x 70); = 40; = 15

Question 80

Find the z value such that 98% of the standard normal curve lies between - z and z.