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Elementary Statistics 11th

Robert Johnson, Patricia Kuby

Chapter 6

Normal Probability Distributions

Educators


Problem 43

Demonstrates that probability is equal to area under a curve. Given that college students sleep an average of 7 hours per night, with a standard deviation equal to 1.7 hours, use the scroll bar in the applet to find:
a. $\quad P(\text { a student sleeps between } 5 \text { and } 9$ hours)
b. $\quad P(\text { a student sleeps between } 2 \text { and } 4$ hours)
c. $\quad P(\text { a student sleeps between } 8 \text { and } 11$ hours)

Gus S.
Numerade Educator

Problem 44

Demonstrates the effects that the mean and standard deviation have on a normal curve.
a. Leaving the standard deviation at $1,$ increase the mean to $3 .$ What happens to the curve?
b. $\quad$ Reset the mean to 0 and increase the standard deviation to $2 .$ What happens to the curve?
c. If you could decrease the standard deviation to 0.5 what do you think would happen to the normal curve?

Gus S.
Numerade Educator

Problem 45

Given $x=58, \mu=43$ and $\sigma=5.2,$ find $z$

Gus S.
Numerade Educator

Problem 46

Given $x=237, \mu=220$ and $\sigma=12.3,$ find $z$.

Gus S.
Numerade Educator

Problem 47

Given that $x$ is a normally distributed random variable with a mean of 60 and a standard deviation of 10 find the following probabilities.
a. $\quad P(x>60)$
b. $\quad P(60< x<72)$
c. $\quad P(57< x<83)$
d. $\quad P(65< x<82)$
e. $\quad P(38< x<78)$
f. $\quad P(x<38)$

Evelyn C.
Numerade Educator

Problem 48

Given that $x$ is a normally distributed random variable with a mean of 28 and a standard deviation of $7,$ find the following probabilities.
a. $\quad P(x<28)$
b. $\quad P(28<x<38)$
c. $\quad P(24<x<40)$
d. $\quad P(30<x<45)$
e. $\quad P(19<x<35)$
f. $\quad P(x<48)$

Evelyn C.
Numerade Educator

Problem 49

As shown in Example $6.8,$ IQ scores are considered normally distributed, with a mean of 100 and a standard deviation of 16.
a. Find the probability that a randomly selected person will have an IQ score between 100 and $120 .$
b. Find the probability that a randomly selected person will have an IQ score above $80 .$

Gus S.
Numerade Educator

Problem 50

Based on a survey conducted by Greenfield Online, 25 to 34-year-olds spend the most each week on fast food. The average weekly amount of $\$ 44$ was reported in a May 2009 USA Today Snapshot. Assuming that weekly fast food expenditures are normally distributed with a standard deviation of $\$ 14.50,$ what is the probability that a 25- to 34-year-old will spend:
a. less than $\$ 25$ a week on fast food?
b. between $\$ 30$ and $\$ 50$ a week on fast food?
c. more than $\$ 75$ a week on fast food?

Gus S.
Numerade Educator

Problem 51

Depending on where you live and on the quality of the day care, costs of day care can range from $\$ 3000$ to $\$ 15,000$ a year (or $\$ 250$ to $\$ 1250$ a month) for one child, according to the Baby Center. Day care centers in large cities such as New York and San Francisco are notoriously expensive. Suppose that day care costs are normally distributed with a mean equal to $\$ 9000$ and a standard deviation equal to $\$ 1800 .$
a. What percentage of day care centers cost between $\$ 7200$ and $\$ 10,800 ?$
b. What percentage of day care centers cost between $\$ 5400$ and $\$ 12,600 ?$
c. What percentage of day care centers cost between $\$ 3600$ and $\$ 14,400 ?$
d. Compare the results in a through c with the empirical rule. Explain the relationship.

Evelyn C.
Numerade Educator

Problem 52

6.52 According to Collegeboard.com [http://www.collegeboard.com/ $]$ the national average salary for a plumber as of 2007 is $\$ 47,350 .$ If we assume that the annual salaries for plumbers are normally distributed with a standard deviation of $\$ 5250,$ find the following:
a. What percentage earn below $\$ 30,000 ?$
b. What percentage earn above $\$ 63,000 ?$

Gus S.
Numerade Educator

Problem 53

According to the Federal Highway Administration's 2006 highway statistics, the distribution of ages for licensed drivers has a mean of 47.5 years and a standard deviation of 16.6 years [www.fhwa.dot.gov]. Assuming the distribution of ages is normally distributed, what percentage of the drivers are:
a. between the ages of 17 and $22 ?$
b. younger than 25 years of age?
c. older than 21 years of age?
d. between the ages of 48 and $68 ?$
e. older than 75 years of age?

Evelyn C.
Numerade Educator

Problem 54

There is a new working class with money to burn, according to the USA Today March 1, 2005, article "New "gold-collar' young workers gain clout." "Gold-collar" is a subset of blue-collar workers, defined by researchers as those working in fast food and retail jobs, or as security guards, office workers, or hairdressers. These 18- to 25-yearold "gold-collar" workers are spending an average of $\$ 729$ a month on themselves (versus $\$ 267$ for college students and $\$ 609$ for blue-collar workers). Assuming this spending is normally distributed with a standard deviation of $\$ 92.00$ what percentage of gold-collar workers spend:
a. between $\$ 600$ and $\$ 900$ a month on themselves?
b. between $\$ 400$ and $\$ 1000$ a month on themselves?
c. more than $\$ 1050$ a month on themselves?
d. less than $\$ 500$ a month on themselves?

Evelyn C.
Numerade Educator

Problem 55

Findings from a survey of American adults conducted by Yankelovich Partners for the International Bottled Water Association indicate that Americans on the average drink 6.18-ounce servings of water a day [http://www.pangaeawater.com/]. Assuming that the number of 8-ounce servings of water is approximately normally distributed with a standard deviation of 1.4 servings, what proportion of Americans drink
a. more than the recommended 8 servings?
b. less than half the recommended 8 servings?

Gus S.
Numerade Educator

Problem 56

As shown in Example $6.12,$ incomes of junior executives are normally distributed with a standard deviation of $\$ 3828.$
a. What is the mean for the salaries of junior executives, if a salary of $\$ 62,900$ is at the top end of the middle $80 \%$ of incomes?
b. With the additional information learned in part a, what is the probability that a randomly selected junior executive earns less than $\$ 50,000 ?$

Gus S.
Numerade Educator

Problem 57

According to ACT, results from the 2008 ACT testing found that students had a mean reading score of 21.4 with a standard deviation of $6.0 .$ Assuming that the scores are normally distributed,
a. find the probability that a randomly selected student had a reading ACT score less than $20 .$
b. find the probability that a randomly selected student had a reading ACT score between 18 and 24
c. find the probability that a randomly selected student had a reading ACT score greater than 30
d. find the value of the 75th percentile for ACT scores.

Gus S.
Numerade Educator

Problem 58

On a given day, the number of square feet of office space available for lease in a small city is a normally distributed random variable with a mean of 750,000 square feet and a standard deviation of 60,000 square feet. The number of square feet available in a second small city is normally distributed with a mean of 800,000 square feet and a standard deviation of 60,000 square feet.
a. Sketch the distribution of leasable office space for both cities on the same graph.
b. What is the probability that the number of square feet available in the first city is less than $800,000 ?$
c. What is the probability that the number of square feet available in the second city is more than $750,000 ?$

Evelyn C.
Numerade Educator

Problem 59

A brewery's filling machine is adjusted to fill quart bottles with a mean of 32.0 oz of ale and a variance of $0.003 .$ Periodically, a bottle is checked and the amount of ale is noted.
a. Assuming the amount of fill is normally distributed, what is the probability that the next randomly checked bottle contains more than 32.02 oz?
b. Let's say you buy 100 quart bottles of this ale for a party; how many bottles would you expect to find containing more than 32.02 oz of ale?

Gus S.
Numerade Educator

Problem 60

Using the standard normal curve and $z$ :
a. Find the minimum score needed to receive an A if the instructor in Example 6.11 said the top $15 \%$ were to get A's.
b. Find the 25 th percentile for IQ scores in Example $6.10 .$
c. If SAT scores are normally distributed with a mean of 500 and a standard deviation of $100,$ what score does a student need to at least be considered by a college that takes only students with scores within the top $7 \% ?$

Gus S.
Numerade Educator

Problem 61

6.61 Final averages are typically approximately normally distributed with a mean of 72 and a standard deviation of
12.5. Your professor says that the top $8 \%$ of the class will receive an $\mathrm{A} ;$ the next $20 \%,$ a $\mathrm{B} ;$ the next $42 \%,$ a $\mathrm{C} ;$ the next $18 \%,$ a $\mathrm{D} ;$ and the bottom $12 \%,$ an $\mathrm{F}$.
a. What average must you exceed to obtain an A?
b. What average must you exceed to receive a grade better than a C?
c. What average must you obtain to pass the course? (You'll need a D or better.)

Gus S.
Numerade Educator

Problem 62

A radar unit is used to measure the speed of automobiles on an expressway during rush-hour traffic. The speeds of individual automobiles are normally distributed with a mean of $62 \mathrm{mph}$.
a. Find the standard deviation of all speeds if $3 \%$ of the automobiles travel faster than $72 \mathrm{mph}$.
b. Using the standard deviation found in part a, find the percentage of these cars that are traveling less than $55 \mathrm{mph}.$
c. Using the standard deviation found in part a, find the 95 th percentile for the variable "speed."

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Problem 63

The weights of ripe watermelons grown at Mr. Smith's farm are normally distributed with a standard deviation of 2.8 lb. Find the mean weight of Mr. Smith's ripe watermelons if only $3 \%$ weigh less than 15 lb.

Gus S.
Numerade Educator

Problem 64

A machine fills containers with a mean weight per container of 16.0 oz. If no more than $5 \%$ of the containers are to weigh less than 15.8 oz, what must the standard deviation of the weights equal? (Assume normality.)

Gus S.
Numerade Educator

Problem 65

"On-hold" times for callers to a local cable television company are known to be normally distributed with a standard deviation of 1.3 minutes. Find the average caller "on-hold" time if the company maintains that no more than $10 \%$ of callers have to wait more than 6 minutes.

Gus S.
Numerade Educator

Problem 66

The data below are the net weights (in grams) for a sample of 30 bags of $\mathrm{M} \& \mathrm{M}$ 's. The advertised net weight is 47.9 grams per bag.
$$\begin{array}{llllll}\hline 46.22 & 46.72 & 46.94 & 47.61 & 47.67 & 47.70 \\47.98 & 48.28 & 48.33 & 48.45 & 48.49 & 48.72 \\48.74 & 48.95 & 48.98 & 49.16 & 49.40 & 49.69 \\49.79 & 49.80 & 49.80 & 50.01 & 50.23 & 50.40 \\50.43 & 50.97 & 51.53 & 51.68 & 51.71 & 52.06 \\\hline\end{array}$$
The FDA requires that (nearly) every bag contain the advertised weight; otherwise, violations (less than 47.9 grams per bag) will bring about mandated fines. (M\&M's are manufactured and distributed by Mars Inc.
a. What percentage of the bags in the sample are in violation?
b. If the weight of all filled bags is normally distributed with a mean weight of 47.9 g, what percentage of the bags will be in violation?
c. Assuming the bag weights are normally distributed with a standard deviation of $1.5 \mathrm{g},$ what mean value would leave $5 \%$ of the weights below $47.9 \mathrm{g} ?$
d. Assuming the bag weights are normally distributed with a standard deviation of $1.0 \mathrm{g},$ what mean value would leave $5 \%$ of the weights below $47.9 \mathrm{g} ?$
e. Assuming the bag weights are normally distributed with a standard deviation of $1.5 \mathrm{g},$ what mean value would leave $1 \%$ of the weights below $47.9 \mathrm{g} ?$
f. Why is it important for Mars to keep the percentage of violations low?
g. It is important for Mars to keep the standard deviation as small as possible so that in turn the mean can be as small as possible to maintain net weight. Explain the relationship between the standard deviation and the mean. Explain why this is important to Mars.

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Problem 67

The extraction force required to remove a cork from a bottle of wine has a normal distribution with a mean of 310 Newtons and a standard deviation of 36 Newtons.
a. The specs for this variable, given in Applied Example $6.13,$ were "$300\mathrm{N}+100 \mathrm{N} /-150 \mathrm{N}$" Express these specs as an interval.
b. What percentage of the corks is expected to fall within the specs?
c. What percentage of the tested corks will have an extraction force of more than 250 Newtons?
d. What percentage of the tested corks will have an extraction force within 50 Newtons of $310 ?$

Evelyn C.
Numerade Educator

Problem 68

The diameter of each cork, as described in Applied Example $6.13,$ is measured in several places and an average diameter is reported for the cork. The average diameter has a normal distribution with a mean of $24.0 \mathrm{mm}$ and standard deviation of $0.13 \mathrm{mm}.$
a. The specs for this variable, given in Applied Example $6.13,$ were $"24 \mathrm{mm}+0.6 \mathrm{mm} /-0.4 \mathrm{mm}"$ Express these specs as an interval.
b. What percentage of the corks is expected to fall within the specs?
c. What percentage of the tested corks will have an average diameter of more than 24.5 millimeters?
d. What percentage of the tested corks will have an average diameter within 0.35 millimeter of 24?

Evelyn C.
Numerade Educator

Problem 69

a. Generate a random sample of 100 data from a normal distribution with mean 50 and standard deviation 12.
b. Using the random sample of 100 data found in part a and the technology commands for calculating ordinate values on page 284, find the 100 corresponding $y$ values for the normal distribution curve with mean 50 and standard deviation 12.
c. Use the 100 ordered pairs found in part b to draw the curve for the normal distribution with mean 50 and standard deviation 12. (Technology commands are included with part b commands on p. 284.)
d. Using the technology commands for cumulative probability on page 285, find the probability that a randomly selected value from a normal distribution with mean 50 and standard deviation 12 will be between 55 and 65. Verify your results by using Table 3.

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Problem 70

Use a computer or calculator to find the probability that one randomly selected value of $x$ from a normal distribution, with mean 584.2 and standard deviation 37.3 will have a value
a. less than 525.
b. between 525 and 590.
c. of at least 590.
d. Verify the result using Table 3.
e. Explain any differences you may find.

Evelyn C.
Numerade Educator

Problem 71

Use a computer to compare a random sample to the population from which the sample was drawn. Consider the normal population with mean 100 and standard deviation 16.
a. List values of $x$ from $\mu-4 \sigma$ to $\mu+4 \sigma$ in increments of half standard deviations and store them in a column.
b. Find the ordinate $(y$ value) corresponding to each abscissa $(x$ value) for the normal distribution curve for $N(100,16)$ and store them in a column.
c. Graph the normal probability distribution curve for $N(100,16).$
d. Generate 100 random data values from the $N(100,16)$ distribution and store them in a column.
e. Graph the histogram of the 100 data obtained in part d using the numbers listed in part a as class boundaries.
f. Calculate other helpful descriptive statistics of the 100 data values and compare the data to the expected distribution. Comment on the similarities and the differences you see.

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Problem 72

Use a computer to compare a random sample to the population from which the sample was drawn. Consider the normal population with mean 75 and standard deviation 14. Answer questions a through $\mathrm{f}$ of Exercise 6.71 using $N(75,14).$

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Problem 73

Suppose you were to generate several random samples, all the same size, all from the same normal probability distribution. Will they all be the same? How will they differ? By how much will they differ?
a. Use a computer or calculator to generate 10 different samples, all of size $100,$ all from the normal probability distribution of mean 200 and standard deviation 25.
b. Draw histograms of all 10 samples using the same class boundaries.
c. Calculate several descriptive statistics for all 10 samples, separately.
d. Comment on the similarities and the differences you see.

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Problem 74

Generate 10 random samples, each of size $25,$ from a normal distribution with mean 75 and standard deviation 14. Answer questions b through d of Exercise 6.73.

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