CHAPTER 5 NORMAL PROBABILITY DISTRIBUTIONS · PDF filesection 5.1 introduction to normal distributions and the standard normal distribution 243 using and interpreting concepts graphical

242 CHAPTER 5 NORMAL PROBABILITY DISTRIBUTIONS

Exercises BUILDING BASIC SKILLS AND VOCABULARY

1. Find three real-life examples of a continuous variable. Which do you thinkmay be normally distributed? Why?

2. In a normal distribution, which is greater, the mean or the median? Explain.

3. What is the total area under the normal curve?

4. What do the inflection points on a normal distribution represent? Where dothey occur?

5. Draw two normal curves that have the same mean but different standarddeviations. Describe the similarities and differences.

6. Draw two normal curves that have different means but the same standarddeviation. Describe the similarities and differences.

7. What is the mean of the standard normal distribution? What is the standarddeviation of the standard normal distribution?

8. Describe how you can transform a nonstandard normal distribution to thestandard normal distribution.

9. Getting at the Concept Why is it correct to say "a" normal distribution and"the" standard normal distribution?

10. Getting at the Concept A z-score is 0. Which of these statements must betrue? Explain your reasoning.

(a) The mean is 0.

(b) The corresponding x-value is 0. •

(c) The corresponding x-value is equal to the mean.

Graphical Analysis In Exercises Il-16, determine whether the graph could

represent a variable with a normal distribution. Explain your reasoning. If the graph

appears to represent a normal distribution, estimate the mean and standard deviation.

11. 12.

JO I I 12 13 14 15 16 17

13. 14.

I 5 I 6 I 7 I 8 I 9 20 2 I 22 45 46 47 48 49 50 5 I 52

15. 16.

12 13 14 15 16 17 18 19 10 I I 12 13 14

Homework

do problems5, 11, 13, 15, 18, 25, 27, 31, 37, 39, 41, 43, 45

SECTION 5.1 INTRODUCTION TO NORMAL DISTRIBUTIONS AND THE STANDARD NORMAL DISTRIBUTION 243

USING AND INTERPRETING CONCEPTS

Graphical Analysis In Exercises 17-22, find the area of the indicated region

under the standard normal curve. If convenient, use technology to find the area.

21.

-1.3 0

0

18.

20.

22.

0 1.7

-2.3 0

Finding Area In Exercises 23-36, find the indicated area under the standard normal curve. lf convenient, use technology to find the area.

23. To the left of z = 0.08

25. To the left of z = -2.575

27. To the right of z = -0.65

29. To the right of z = -0.355

31. Between z = 0 and z = 2.86

33. Between z = -1.96 and z = 1.96

34. Between z = -2.33 and z = 2.33

24. To the left of z = -3.16

26. To the left of z = 1.365

28. To the right of z = 3.25

30. To the right of z = l.615

32. Between z = -1.53 and z = 0

35. To the left of z = -1.28 and to the right of z = 1.28

36. To the left of z = -1.96 and to the right of z = 1.96

37. Manufacturer Claims You work for a consumer watchdog publication and

are testing the advertising claims of a tire manufacturer. The manufacturerclaims that the life spans of the tires are normally distributed, with a meanof 40,000 miles and a standard deviation of 4000 miles. You test 16 tires and

record the life spans shown below.

48,778 41,046 29,083 36,394 32,302 42,787 41,972 37,229 25,314 31,920 38,030 38,445 30,750 38,886 36,770 46,049

(a) Draw a frequency histogram to display these data. Use five classes.Do the life spans appear to be normally distributed? Explain.

(b) Find the mean and standard deviation of your sample.

(c) Compare the mean and standard deviation of your sample with thosein the manufacturer's claim. Discuss the differences.


38. Milk Consumption You are performing a study about weekly percapita milk consumption. A previous study found weekly per capitamilk consumption to be normally distributed, with a mean of48.7 fluid ounces and a standard deviation of 8.6 fluid ounces. Yourandomly sample 30 people and record the weekly milk consumptionsshown below.

40 45 54 41 43 31 47 30 33 37 48 57 52 45 38 65 25 39 53 51 58 52 40 46 44 48 61 47 49 57

(a) Draw a frequency histogram to display these data. Use seven classes.Do the consumptions appear to be normally distributed? Explain.

(b) Find the mean and standard deviation of your sample.

(c) Compare the mean and standard deviation of your sample withthose of the previous study. Discuss the differences.

Computing and Interpreting z-Scores In Exercises 39 and 40, (a) find the z-scores that corresponds to each value and (b) determine whether any of the values are unusual.

39. SAT Scores The SAT is an exam used by colleges and universities to evaluate undergraduate applicants. The test scores are normally distributed.In a recent year, the mean test score was 1498 and the standard deviationwas 316. The test scores of four students selected at random are 1920, 1240,2200, and 1390.

40. ACT Scores The ACT is an exam used by colleges and universities toevaluate undergraduate applicants. The test scores are normally distributed.In a recent year, the mean test score was 2L1 and the standard deviationwas 5.3. The test scores of four students selected at random are 15, 22, 9,and 35.

Graphical Analysis In Exercises 41-46, find the probability of z occurring in the indicated region of the slandard normal dis1ribu1ion. ff convenienl, use technology to find the probabilily.

41.

43.

-2.005

45.

0

0

-I O

42.

1.96

44.

46.

-0.875 0

0 1.28

0 1.54

SECTION 5.1 INTRODUCTION TO NORMAL DISTRIBUTIONS AND THE STANDARD NORMAL DISTRIBUTION 245

• I I I I I I I I I I I•

Finding Probabilities In Exercises 47-56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

47. P(z < 1.45)

50. P(z > -1.85)

48. P( z < -0.18) 49. P(z > 2.175)

51. P(-0.89 < z < 0) 52. P(O < z < 0.525)

53. P(-1.65 < z < 1.65) 54. P(-1.54 < z < 1.54)

55. P(z < -2.58 or z > 2.58) 56. P(z < -1.54 or z > 1.54)

EXTENDING CONCEPTS

57. Writing Draw a normal curve with a mean of 60 and a standard deviationof 12. Describe how you constructed the curve and discuss its features.

58. Writing Draw a normal curve with a mean of 450 and a standard deviationof 50. Describe how you constructed the curve and discuss its features.

Uniform Distribution A uniform distribution is a continuous probability distribution for a random variable x between two values a and b ( a < b), where a s x s b and all of the values of x are equally likely to occur. The graph of a uniform distribution is shown below.

I

b-a

y

{I b

The probability density function of a uniform distribution is

1 y =

b-a

on the interval from x = a to x = b. For any value of x less than a or greater than b, y = 0. In Exercises 59 and 60, use this information.

59. Show that the probability density function of a uniform distribution satisfiesthe two conditions for a probability density function.

60. For two values c and d, where a s c < d s b, the probability that x lies between c and dis equal to the area under the curve between c and d, as shown below.

I

b-{I

y

a d b

So, the area of the red region equals the probability that x lies between c and d.

For a uniform distribution from a = 1 to b = 25, find the probability that

(a) x lies between 2 and 8.

(b) x lies between 4 and 12.

(c) x lies between 5 and 17.

(d) x lies between 8 and 14.

Exercises

SECTION 5.2 NORMAL DISTRIBUTIONS: FINDING PROBABILITIES 249

BUILDING BASIC SKILLS AND VOCABULARY

Computing Probabilities In Exercises 1-6, the random variable x is normally distributed with meanµ, = 174 and standard deviation <r = 20. Find the indicated probability.

1. P(x < 170)

3. P(x > 182)

5. P(l60 < x < 170)

2. P(x < 200)

4. P(x > 155)

6. P(l72 < x < 192)


Finding Probabilities In Exercises 7-12, find the indicated probabilities. If convenient, use technology to find the probabilities.

7. Heights of Men In a survey of U.S. men, the heights in the 20-29age group were normally distributed, with a mean of 69.4 inches and astandard deviation of 2.9 inches. Find the probability that a randomlyselected study participant has a height that is (a) less than 66 inches,(b) between 66 and 72 inches, and (c) more than 72 inches, and (d) identifyany unusual events. Explain your reasoning.

8. Heights of Women ln a survey of U.S. women, the heights in the 20-29 agegroup were normally distributed, with a mean of 64.2 inches and a standarddeviation of 2.9 inches. Find the probability that a randomly selected studyparticipant has a height that is (a) Jess than 56.5 inches, (b) between 61 and67 inches, and (c) more than 70.5 inches, and (d) identify any unusual events.Explain your reasoning.

9. ACT Reading Scores In a recent year, the ACT scores for the readingportion of the test were normally distributed, with a mean of 21.3 and astandard deviation of 6.2. Find the probability that a randomly selected highschool student who took the reading portion of the ACT has a score that is(a) less than 15, (b) between 18 and 25, and (c) more than 34, and (d) identifyany unusual events. Explain your reasoning.

10. ACT Math Scores In a recent year, the ACT scores for the math portionof the test were normally distributed, with a mean of 21.1 and a standarddeviation of 5.3. Find the probability that a randomly selected high schoolstudent who took the math portion of the ACT has a score that is (a) lessthan 16, (b) between 19 and 24, and (c) more than 26, and (d) identify anyunusual events. Explain your reasoning.

11. Utility Bills The monthly utility bills in a city are normally distributed, witha mean of $100 and a standard deviation of $12. Find the probability that arandomly selected utility bill is (a) less than $70, (b) between $90 and $120,and (c) more than $140.

12. Health Club Schedule The amounts of time per workout an athlete usesa stairclimber are normally distributed, with a mean of 20 minutes anda standard deviation of 5 minutes. Find the probability that a randomlyselected athlete uses a stairclimber for (a) less than 17 minutes, (b) between20 and 28 minutes, and (c) more than 30 minutes.

Homework

do problems1-19 ODD


Graphical Analysis In Exercises 13-16, a member is selected at random from the population represented by the graph. Find the probability that the member selected at random is from the shaded area of the graph. Assume the variable x is normally distributed.

15.

SAT Writing Scores

200 450

Score

µ=488

<J= 114

800

U.S. Men Ages 35-44:

Total Cholesterol

µ= 205

<J=37.8

..,"""":...._��---4�+-'""""�.·� 75 220 255 300

14.

16.

SAT Math Scores

µ= 514

<J= 117

200 670 800

Score

U.S. Women Ages 35-44:

100

Total Cholesterol

190 '215

µ= 195

<J= 37.7

300

Using Normal Distributions In Exercises 17-20, answer the questions about the specified normal distribution.

17. SAT Writing Scores Use the normal distribution in Exercise 13.

(a) What percent of the SAT writing scores are less than 600?

(b) Out of 1000 randomly selected SAT writing scores, about how many

would you expect to be greater than 500?

18. SAT Math Scores Use the normal distribution in Exercise 14.

(a) What percent of the SAT math scores are less than 500?

(b) Out of 1500 randomly selected SAT math scores, about how manywould you expect to be greater than 600?

19. Cholesterol Use the normal distribution in Exercise 15.

(a) What percent of the men have a total cholesterol level less than

225 milligrams per deciliter of blood?

(b) Out of 250 randomly selected U.S. men in the 35-44 age group, about

how many would you expect to have a total cholesterol level greaterthan 260 milligrams per deciliter of blood?

20. Cholesterol Use the normal distribution in Exercise 16.

(a) What percent of the women have a total cholesterol level less than217 milligrams per deciliter of blood?

(b) Out of 200 randomly selected U.S. women in the 35-44 age group, about

how many would you expect to have a total cholesterol level greaterthan 185 milligrams per deciliter of blood?

Exercises

SECTION 5.3 NORMAL DISTRIBUTIONS: FINDING VALUES 257

BUILDING BASIC SKILLS AND VOCABULARY

In Exercises 1-16, use the Standard Normal Table lo find the z-score that corresponds to the cumulative area or percentile. If the area is not in the table, use the entry closest to the area. If the area is halfway between two entries, use the z-score halfway between the corresponding z-scores. If convenient, use technologyto find the z-score.

1. 0.2090

5. 0.05

2. 0.4364

6. 0.85

10. P30

14. P40

3. 0.9916

7. 0.94

11. Pss

15. P75

4. 0.7995

8. 0.0046

12. p67

16. P80

Graphical Analysis In Exercises 17-22, find the indicated z-score(s) shown in the graph. If convenient, use technology to find the z-score(s).

17. 18.

19. 20.

0 z=?

21. 22.

z=? 0 z=? z=? 0 z=?

In Exercises 23-30, find the indicated z-score.

23. Find the z-score that has 11.9% of the distribution's area to its left.

24. Find the z-score that has 78.5% of the distribution's area to its left.

25. Find the z-score that has 11.9% of the distribution's area to its right.

26. Find the z-score that has 78.5% of the distribution's area to its right.

27. Find the z-score for which 80% of the distribution's area lies between -zand z.

28. Find the z-score for which 99% of the distribution's area lies between -zand z.

Homework

do

problems1, 5, 9, 13, 17, 21, 23, 27, 31, 33, 35, 37


29. Find the z-score for which 5% of the distribution's area lies between -z

and z.

30. Find the z-score for which 12 % of the distribution's area lies between -zand z.


• Using Normal Distributions In Exercises 31-38, answer the questions about

{9 � f@ }'.'::::::;:.:::":":::my of womeo io the Uoited State, (ages 20-29),

\... _ � the mean height was 64.2 inches with a standard deviation of 2.9 inches.

Sleeping Times of

Medical Residents

4 6 7

Hours

9

FIGURE FOR EXERCISE 35

� (a) What height represents the 95th percentile? (b) What height represents the first quartile?

32. Heights of Men In a survey of men in the United States (ages 20-29),the mean height was 69.4 inches with a standard deviation of 2.9 inches.

(a) What height represents the 90th percentile?(b) What height represents the first quartile?

33. Heart Transplant Waiting Times The time spent (in days) waiting for aheart transplant for people ages 35-49 can be approximated by a normaldistribution, as shown in the figure.

(a) What waiting time represents the 5th percentile?(b) What waiting time represents the third quartile? •

Time Spent Waiting for a Heart

µ= 203 days

CY= 25.7 days

I 00 I 50 200 250 300

Days


Time Spent Waiting for a Kidney

I 200 I 600 2000 2400

Days


34. Kidney Transplant Waiting Times The time spent (in days) waiting for akidney transplant for people ages 35-49 can be approximated by a normaldistribution, as shown in the figure.

(a) What waiting time represents the 80th percentile?(b) What waiting time represents the first quartile?

35. Sleeping Times of Medical Residents The average time spent sleeping (inhours) for a group of medical residents at a hospital can be approximatedby a normal distribution, as shown in the figure.

(a) What is the shortest time spent sleeping that would still place a residentin the top 5% of sleeping times?

(b) Between what two values does the middle 50% of the sleep times lie?

Annual U.S. per Capita

lee Cream Consumption

4 8 12 16 20 24 28 32

Consump1ion (in pounds)


Final Exam Grades

D C B A

Poims scored on final exam


SECTION 5.3 NORMAL DISTRIBUTIONS: FINDING VALUES 259

36. Ice Cream The annual per capita consumption of ice cream (in pounds) inthe United States can be approximated by a normal distribution, as shown inthe figure.

(a) What is the largest annual per capita consumption of ice cream that canbe in the bottom 10% of consumptions?

(b) Between what two values does the middle 80% of the consumptions lie?

37. Apples The annual per capita consumption of fresh apples (in pounds)in the United States can be approximated by a normal distribution, with amean of 9.5 pounds and a standard deviation of 2.8 pounds.

38.

(a) What is the smallest annual per capita consumption of apples that can bein the top 25% of consumptions?

(b) What is the largest annual per capita consumption of apples that can bein the bottom 15% of consumptions?

Bananas The annual per capita consumption of fresh bananas (in pounds) in the United States can be approximated by a normal distribution, with a mean of 10.4 pounds and a standard deviation of 3 pounds.

(a) What is the smallest annual per capita consumption of bananas that canbe in the top 10% of consumptions?

(b) What is the largest annual per capita consumption of bananas that can

be in the bottom 5% of consumptions?

39. Bags of Baby Carrots The weights of bags of baby carrots are normallydistributed, with a mean of 32 ounces and a standard deviation of 0.36 ounce.Bags in the upper 4.5% are too heavy and must be repackaged. What is themost a bag of baby carrots can weigh and not need to be repackaged?

40. Writing a Guarantee You sell a brand of automobjle tire that has a lifeexpectancy that is normally distributed, with a mean life of 30,000 milesand a standard deviation of 2500 miles. You want to give a guaranteefor free replacement of tires that do not wear well. You are willing

to replace approximately 10% of the tires. How should you word

your guarantee?

EXTENDING CONCEPTS

41. Vending Machine A vending machine dispenses coffee into an eight-ouncecup. The amounts of coffee dispensed into the cup are normally distributed,with a standard deviation of 0.03 ounce. You can allow the cup to overflow

1 % of the time. What amount should you set as the mean amount of coffeeto be dispensed?

42. Statistics Grades In a large section of a statistics class, the points for thefinal exam are normally distributed, with a mean of 72 and a standarddeviation of 9. Grades are assigned according to the following rule.

• The top 10% receive A's.

• The next 20% receive B's.

• The middle 40% receive C's.

• The next 20% receive D's.

• The bottom 10% receive F's.

Find the lowest score on the final exam that would qualify a student for an A, a B, a C, and a D.


10. The annual snowfall (in feet) for a central New York state county

P(x)

;'.;' 5 0.12

t!: 0.08

-� 'ii 0.04 "il

(a)

/

4 6 10

Snowfall (in feet)

<>:t = 0.23 (b) µ

x = 0.58

-0.5 0 0.5 1.0 1.5

Snowfall (in feet)

cr:r

=0.23 (c)

µx = 5.8

-+-+-'-""""-+---+---x

2 4 6 8 10

Snowfall (in feet)

-2 0 2 4 6 8 10 12

Snowfall (in feet)

Verifying Properties of Sampling Distributions In Exercises Il-14,

find the mean and standard deviation of the population. List all samples (with replacement) of the given size from Lhat populalion and find the mean of

each. Find the mean and standard deviation of the sampling distribulion of sample means and compare them with the mean and standard devialion of the population.

11. The word counts of five essays are 501, 636, 546, 602, and 575. Use a samplesize of 2.

12. The amounts four friends paid for their MP3 players are $200, $130, $270,and $230. Use a sample size of 2.

13. The scores of three students in a study group on a test are 98, 95, and 93.Use a sample size of 3.

14. The numbers of DVDs rented by each of four families in the past month are8, 4, 16, and 2. Use a sample size of 3.

Finding Probabilities In Exercises 15-18, the population mean and standard deviation are given. Find the indicated probability and determine whether the

given sample mean would be considered unusual. If convenienl, use Lechnology Lo find Lhe probability.

15. For a sample of n = 64, find the probability of a sample mean being less than24.3 when µ = 24 and er = 1.25.

16. For a sample of n = 100, find the probability of a sample mean being greaterthan 24.3 when µ = 24 and er = 1.25.

17. For a sample of n = 45, find the probability of a sample mean being greaterthan 551 when µ = 550 and er = 3.7.

18. For a sample of n = 36, find the probability of a sample mean being less than12,750 or greater than 12,753 when µ = 12,750 and er = 1.7.

SECTION 5.4 SAMPLING DISTRIBUTIONS AND THE CENTRAL LIMIT THEOREM 271


Using the Central Limit Theorem In Exercises 19-24, use the Central

Limit Theorem to find the mean and standard deviation of the indicated sampling

distribution of sample means. Then sketch a graph of the sampling distribution.

19. Braking Distances The braking distances (from 60 miles per hour to acomplete stop on dry pavement) of a sports utility vehicle are normallydistributed, with a mean of 154 feet and a standard deviation of 5.12 feet.Random samples of size 12 are drawn from this population, and the mean ofeach sample is determined.

20. Braking Distances The braking distances (from 60 miles per hour to acomplete stop on dry pavement) of a car are normally distributed, with amean of 136 feet and a standard deviation of 4.66 feet. Random samplesof size 15 are drawn from this population, and the mean of each sample isdetermined.

21. SAT Critical Reading Scores: Males The scores for males on the critical reading portion of the SAT are normally distributed, with a mean of 498 and a standarddeviation of 116. Random samples of size 20 are drawn from this population,and the mean of each sample is determined.

22. SAT Critical Reading Scores: Females The scores for females on thecritical reading portion of the SAT are normally distributed, with a mean of493 and a standard deviation of 112. Random samples of size 36 are drawnfrom this population, and the mean of each sample is determined.

23. Canned Fruit The annual per capita consumption of canned fruit by peoplein the United States is normally distributed, with a mean of 10 pounds and astandard deviation of 1.8 pounds. Random samples of size 25 are drawn fromthis population, and the mean of each sample is determined.

24. Canned Vegetables The annual per capita consumption of cannedvegetables by people in the United States is normally distributed, with amean of 39 pounds and a standard deviation of 3.2 pounds. Random samplesof size 30 are drawn from this population, and the mean of each sample isdetermined.

25. Repeat Exercise 19 for samples of size 24 and 36. What happens to the meanand the standard deviation of the distribution of sample means as the size ofthe sample increases?

26. Repeat Exercise 20 for samples of size 30 and 45. What happens to the meanand the standard deviation of the distribution of sample means as the size ofthe sample increases?

Finding Probabilities In Exercises 27-32, find the indicated probability and interpret the results. Tf convenient, use technology to find Lhe probabilily.

27. Salaries The mean annual salary for environmental compliance specialistsis about $66,000. A random sample of 35 specialists is selected from thispopulation. What is the probability that the mean salary of the sample is lessthan $60,000? Assume <r = $12,000.

28. Salaries The mean annual salary for flight attendants is about $65,700.A random sample of 48 flight attendants is selected from this population.What is the probability that the mean annual salary of the sample is less than$63,400? Assume <r = $14,500.

homework

do

problems

15-37 ODD

Section 5.4


29.

32.

33.

34.

Gas Prices: New England During a certain week, the mean price of gasoline in the New England region was $3.796 per gallon. A random sample of 32 gas stations is selected from this population. What is the probability that the mean price for the sample was between $3.781 and $3.811 that week? Assume a = $0.045.

Gas Prices: California During a certain week, the mean price of gasoline in California was $4.117 per gallon. A random sample of 38 gas stations is selected from this population. What is the probability that the mean price for the sample was between $4.128 and $4.143 that week? Assume a = $0.049.

Heights of Women The mean height of women in the United States (ages 20-29) is 64.2 inches. A random sample of 60 women in this age group isselected. What is the probability that the mean height for the sample isgreater than 66 inches? Assume a = 2.9 inches.

Heights of Men The mean height of men in the United States (ages 20-29) is 69.4 inches. A random sample of 60 men in this age group is selected. What is the probability that the mean height for the sample is greater than 70 inches? Assume a = 2.9 inches.

Which Is More Likely? Assume that the heights in Exercise 31 are normally distributed. Are you more likely to randomly select l woman with a height less than 70 inches or are you more likely to select a sample of 20 women with a mean height less than 70 inches? Explain.

Which Is More Likely? Assume that the heights in Exercise 32 are normally distributed. Are you more likely to randomly select 1 man with a height less than 65 inches or are you more likely to select a sample of 15 men with a mean height less than 65 inches? Explain.

35. Paint Cans A machine is set to fill paint cans with a mean of 128 ouncesand a standard deviation of 0.2 ounce. A random sample of 40 cans has amean of 127.9 ounces. Does the machine need to be reset? Explain.

36. Milk Containers A machine is set to fill milk containers with a mean of64 ounces and a standard deviation of 0.11 ounce. A random sample of40 containers has a mean of 64.05 ounces. Does the machine need to bereset? Explain.

37. Lumber Cutter The lengths of lumber a machine cuts are normallydistributed, with a mean of 96 inches and a standard deviation of 0.5 inch.

(a) What is the probability that a randomly selected board cut by themachine has a length greater than 96.25 inches?

(b) You randomly select 40 boards. What is the probability that their meanlength is greater than 96.25 inches?

(c) Compare the probabilities from parts (a) and (b).

38. Ice Cream The weights of ice cream cartons produced by a manufacturerare normally distributed with a mean weight of 10 ounces and a standarddeviation of 0.5 ounce.

(a) What is the probability that a randomly selected carton has a weightgreater than 10.21 ounces?

(b) You randomly select 25 cartons. What is the probability that their meanweight is greater than 10.21 ounces?

(c) Compare the probabilities from parts (a) and (b).

Documents

CHAPTER 5 NORMAL PROBABILITY DISTRIBUTIONS · PDF filesection 5.1 introduction to normal distributions and the standard normal distribution 243 using and interpreting concepts graphical