Chapter 6, The Normal probability Distribution Video Solutions, Introduction to Probability and Statistics | Numerade (2024)

Consider a binomial random varible with $n=25$ and $p=.6 .$ Fill in the blanks below to find some probabilities using the normal approximation.
a. Can we use the normal approximation? Calculate $n p=$ _____ and $n q=$ _____
b. Are $n p$ and $n q$ both greater than $5 ?$ Yes ____ No ____
c. If the answer to part $b$ is yes, calculate $\mu=n p=$ ______ and $\sigma=\sqrt{n p q}=$ ______
d. To find the probability of more than 9 successes, what values of $x$ should be included? $x=$ ________
e. To include the entire block of probability for the first value of $x=$ ______, start at _______.
f. Calculate $z=\frac{x \pm .5-n p}{\sqrt{n p q}}=$ _______.
g. Calculate $P(x>9) \approx P(z>$______) $=1-$ _____ $=$ ____.

Consider a binomial random variable with $n=45$ and $p=.05 .$ Fill in the blanks below to find some probabilities using the normal approximation.
a. Can we use the normal approximation? Calculate $n p=$ ______ and $n q=$ ______
b. Are $n p$ and $n q$ both greater than $5 ?$ Yes_____ No______
c. If the answer to part $b$ is yes, calculate $\mu=n p=$ _______ and $\sigma=\sqrt{n p q}=$ _____
d. To find the probability of 10 or fewer successes, what values of $x$ should be included? $x=$ _______
e. To include the entire block of probability for the first value of $x=$ ______, start at ______.
f. Calculate $z=\frac{x \pm .5-n p}{\sqrt{n p q}}=$ _______.
g. Calculate $P(x \leq 10) \approx P(z<$_____ ) $=$ ______.

Let $x$ be a binomial random variable with $n=25$ and $p=.3$
a. Is the normal approximation appropriate for this binomial random variable?
b. Find the mean and standard deviation for $x$.
c. Use the normal approximation to find $P(6 \leq x \leq 9)$.
d. Use Table 1 in Appendix I to find the exact probability $P(6 \leq x \leq 9)$. Compare the results of parts $c$ and d. How close was your approximation?

Let $x$ be a binomial random variable with $n=15$ and $p=.5$
a. Is the normal approximation appropriate?
b. Find $P(x \geq 6)$ using the normal approximation.
c. Find $P(x>6)$ using the normal approximation.
d. Find the exact probabilities for parts $b$ and $c,$ and compare these with your approximations.

Let $x$ be a binomial random variable with $n=100$ and $p=.2 .$ Find approximations to these probabilities:
a. $P(x>22)$
b. $P(x \geq 22)$
c. $P(20<x<25)$
d. $P(x \leq 25)$

Let $x$ be a binomial random variable for $n=25,$ $p=.2$
a. Use Table 1 in Appendix I to calculate $P(4 \leq x \leq 6)$.
b. Find $\mu$ and $\sigma$ for the binomial probability distribution, and use the normal distribution to approximate the probability $P(4 \leq x \leq 6)$. Note that this value is a good approximation to the exact value of $P(4 \leq x \leq 6)$ even though $n p=5$

Suppose the random variable $x$ has a binomial distribution corresponding to $n=20$ and $p=.30 .$ Use Table 1 of Appendix I to calculate these probabilities:
a. $P(x=5)$
b. $P(x \geq 7)$

Refer to Exercise $6.41 .$ Use the normal approximation to calculate $P(x=5)$ and $P(x \geq 7)$. Compare with the exact values obtained from Table 1 in
Appendix I.

Consider a binomial experiment with $n=20$ and $p=.4 .$ Calculate $P(x \geq 10)$ using each of these methods:
a. Table 1 in Appendix I
b. The normal approximation to the binomial probability distribution

Find the normal approximation to $P(355 \leq x \leq 360)$ for a binomial probability distribution with $n=400$ and $p=.9$.

How often do you watch movies at home? A USA Today Snapshot found that about 7 in 10 adults say they watch movies at home at least once a week. $^{5}$ Suppose a random sample of $n=50$ adults are polled and asked if they had watched a movie at home this week. Let us assume that $p=.7$ is, in fact, correct. What are the probabilities for the following events?
a. Fewer than 30 individuals watched a movie at home this week?
b. More than 42 individuals watched a movie at home this week?
c. Fewer than 10 individuals did not watch a movie at home this week?

Data collected over a long period of time show that a particular genetic defect occurs in 1 of every 1000 children. The records of a medical clinic show $x=60$ children with the defect in a total of 50,000 examined. If the 50,000 children were a random sample from the population of children represented by past records, what is the probability of observing a value of $x$ equal to 60 or more? Would you say that the observation of $x=60$ children with genetic defects represents a rare event?

Airlines and hotels often grant reservations in excess of capacity to minimize losses due to no-shows. Suppose the records of a hotel show that, on the average, $10 \%$ of their prospective guests will not claim their reservation. If the hotel accepts 215 reservations and there are only 200 rooms in the hotel, what is the probability that all guests who arrive to claim a room will receive one?

Compilation of large masses of data on lung cancer shows that approximately 1 of every 40 adults acquires the disease. Workers in a certain occupation are known to work in an air-polluted environment that may cause an increased rate of lung cancer. A random sample of $n=400$ workers shows 19 with identifiable cases of lung cancer. Do the data provide sufficient evidence to indicate a higher rate of lung cancer for these workers than for the national average?

Is a tall president better than a short one? Do Americans tend to vote for the taller of the two candidates in a presidential selection? In 33 of our presidential elections between 1856 and $2006,$ 17 of the winners were taller than their opponents. Assume that Americans are not biased by a candidate's height and that the winner is just as likely to be taller or shorter than his opponent. Is the observed number of taller winners in the U.S. presidential elections unusual?
a. Find the approximate probability of finding 17 or more of the 33 pairs in which the taller candidate wins.
b. Based on your answer to part a, can you conclude that Americans might consider a candidate's height when casting their ballot?

In a certain population, $15 \%$ of the people have Rh-negative blood. A blood bank serving this population receives 92 blood donors on a particular day.
a. What is the probability that 10 or fewer are Rh-negative?
b. What is the probability that 15 to 20 (inclusive) of the donors are Rh-negative?
c. What is the probability that more than 80 of the donors are Rh-positive?

Two of the biggest soft drink rivals, Pepsi and co*ke, are very concerned about their market share. The following pie chart, which appeared on the company website (http:// www.pepsico.com) in November, 2006 , claims that Pepsi-Cola's share of the U.S. beverage market is $26 \% .^{6}$ Assume that this proportion will be close to the probability that a person selected at random indicates a preference for a Pepsi product when choosing a soft drink.
a. Exactly 150 consumers prefer a Pepsi product.
b. Between 120 and 150 consumers (inclusive) prefer a Pepsi product.
c. Fewer than 150 consumers prefer a Pepsi product.
d. Would it be unusual to find that 232 of the 500 consumers preferred a Pepsi product? If this were to occur, what conclusions would you draw?

The typical American family spends lots of time driving to and from various activities, and lots of time in the drive-thru lines at fast-food restaurants. There is a rising amount of evidence suggesting that we are beginning to burn out! In fact, in a study conducted for the Center for a New American Dream, Time magazine reports that $60 \%$ of Americans felt pressure to work too much, and $80 \%$ wished for more family time. ${ }^{7}$ Assume that these percentages are correct for all Americans, and that a random sample of 25 Americans is selected.
a. Use Table 1 in Appendix I to find the probability that more than 20 felt pressure to work too much.
b. Use the normal approximation to the binomial distribution to aproximate the probability in part a. Compare your answer with the exact value from part a.
c. Use Table 1 in Appendix I to find the probability that between 15 and 20 (inclusive) wished for more family time.
d. Use the normal approximation to the binomial distribution to approximate the probability in part c. Compare your answer with the exact value from
part c.

The article in Time magazine $^{7}$ (Exercise 6.52 ) also reported that $80 \%$ of men and $62 \%$ of women put in more than 40 hours a week on the job. Assume that these percentages are correct for all Americans, and that a random sample of 50 working women is selected.
a. What is the average number of women who put in more than 40 hours a week on the job?
b. What is the standard deviation for the number of women who put in more than 40 hours a week on the job?
c. Suppose that in our sample of 50 working women, there are 25 who work more than 40 hours a week. Would you consider this to be an unusual occurrence? Explain.

Using Table 3 in Appendix I, calculate the area under the standard normal curve to the left of the following:
a. $z=1.2$
b. $z=-.9$
c. $z=1.46$
d. $z=-.42$

Find the following probabilities for the standard normal random variable:
a. $P(.3<z<1.56)$
b. $P(-.2<z<.2)$

a. Find the probability that $z$ is greater than $-.75 .$
b. Find the probability that $z$ is less than 1.35 .

Find $z_{0}$ such that $P\left(z>z_{0}\right)=.5$

Find the probability that $z$ lies between $z=-1.48$ and $z=1.48$.

Find $z_{0}$ such that $P\left(-z_{0}<z<z_{0}\right)=.5 .$ What percentiles do $-z_{0}$ and $z_{0}$ represent?

The life span of oil-drilling bits depends on the types of rock and soil that the drill encounters, but it is estimated that the mean length of life is 75 hours. Suppose an oil exploration company purchases drill bits that have a life span that is approximately normally distributed with a mean equal to 75 hours and a standard deviation equal to 12 hours.
a. What proportion of the company's drill bits will fail before 60 hours of use?
b. What proportion will last at least 60 hours?
c. What proportion will have to be replaced after more than 90 hours of use?

The influx of new ideas into a college or university, introduced primarily by new young faculty, is becoming a matter of concern because of the increasing ages of faculty members; that is, the distribution of faculty ages is shifting upward due most likely to a shortage of vacant positions and an oversupply of PhDs. Thus, faculty members are reluctant to move and give up a secure position. If the retirement age at most universities is $65,$ would you expect the distribution of faculty ages to be normal? Explain.

A machine operation produces bearings whose diameters are normally distributed, with mean and standard deviation equal to .498 and .002, respectively. If specifications require that the bearing diameter equal .500 inch ±.004 inch, what fraction of the production will be unacceptable?

A used-car dealership has found that the length of time before a major repair is required on the cars it sells is normally distributed with a mean equal to 10 months and a standard deviation of 3 months. If the dealer wants only $5 \%$ of the cars to fail before the end of the guarantee period, for how many months should the cars be guaranteed?

The daily sales total (excepting Saturday) at a small restaurant has a probability distribution that is approximately normal, with a mean $\mu$ equal to $\$ 1230$ per day and a standard deviation $\sigma$ equal to $\$ 120$.
a. What is the probability that the sales will exceed $\$ 1400$ for a given day?
b. The restaurant must have at least $\$ 1000$ in sales per day to break even. What is the probability that on a given day the restaurant will not break even?

The life span of a type of automatic washer is approximately normally distributed with mean and standard deviation equal to 10.5 and 3.0 years, respectively. If this type of washer is guaranteed for a period of 5 years, what fraction will need to be repaired and/or replaced?

Most users of automatic garage door openers activate their openers at distances that are normally distributed with a mean of 30 feet and a standard deviation of 11 feet. To minimize interference with other remote-controlled devices, the manufacturer is required to limit the operating distance to 50 feet. What percentage of the time will users attempt to operate the opener outside its operating limit?

The average length of time required to complete a college achievement test was found to equal 70 minutes with a standard deviation of 12 minutes. When should the test be terminated if you wish to allow sufficient time for $90 \%$ of the students to complete the test? (Assume that the time required to complete the test is normally distributed.)

The length of time required for the periodic maintenance of an automobile will usually have a probability distribution that is mound-shaped and, because some long service times will occur occasionally, is skewed to the right. The length of time required to run a 5000 -mile check and to service an automobile has a mean equal to 1.4 hours and a standard deviation of .7 hour. Suppose that the service department plans to service 50 automobiles per 8 -hour day and that, in order to do so, it must spend no more than an average of 1.6 hours per automobile. What proportion of all days will the service department have to work overtime?

An advertising agency has stated that $20 \%$ of all television viewers watch a particular program. In a random sample of 1000 viewers, $x=184$ viewers were watching the program. Do these data present sufficient evidence to contradict the advertiser's claim?

A researcher notes that senior corporation executives are not very accurate forecasters of their own annual earnings. He states that his studies of a large number of company executive forecasts "showed that the average estimate missed the mark by $15 \%$."
a. Suppose the distribution of these forecast errors has a mean of $15 \%$ and a standard deviation of $10 \%$. Is it likely that the distribution of forecast errors is approximately normal?
b. Suppose the probability is .5 that a corporate executive's forecast error exceeds $15 \% .$ If you were to sample the forecasts of 100 corporate executives, what is the probability that more than 60 would be in error by more than $15 \% ?$

A soft drink machine can be regulated to discharge an average of $\mu$ ounces per cup. If the ounces of fill are normally distributed, with standard deviation equal to .3 ounce, give the setting for $\mu$ so that 8 -ounce cups will overflow only $1 \%$ of the time.

A manufacturing plant uses 3000 electric light bulbs whose life spans are normally distributed, with mean and standard deviation equal to 500 and 50 hours, respectively. In order to minimize the number of bulbs that burn out during operating hours, all the bulbs are replaced after a given period of operation. How often should the bulbs be replaced if we wish no more than $1 \%$ of the bulbs to burn out
between replacement periods?

The admissions office of a small college is asked to accept deposits from a number of qualified prospective freshmen so that, with probability about $.95,$ the size of the freshman class will be less than or equal to $120 .$ Suppose the applicants constitute a random sample from a population of applicants, $80 \%$ of whom would actually enter the freshman class if accepted.
a. How many deposits should the admissions counselor accept?
b. If applicants in the number determined in part a are accepted, what is the probability that the freshman class size will be less than $105 ?$

An airline finds that $5 \%$ of the persons making reservations on a certain flight will not show up for the flight. If the airline sells 160 tickets for a flight that has only 155 seats, what is the probability that a seat will be available for every person holding a reservation and planning to fly?

It is known that $30 \%$ of all calls coming into a telephone exchange are long-distance calls. If 200 calls come into the exchange, what is the probability that at least 50 will be long-distance calls?

In Exercise $5.75,$ a cross between two peony plants-one with red petals and one with streaky petals-produced offspring plants with red petals $75 \%$ of the time. Suppose that 100 seeds from this cross were collected and germinated, and $x$, the number of plants with red petals, was recorded.
a. What is the exact probability distribution for $x ?$
b. Is it appropriate to approximate the distribution in part a using the normal distribution? Explain.
c. Use an appropriate method to find the approximate probability that between 70 and 80 (inclusive) offspring plants have red flowers.
d. What is the probability that 53 or fewer offspring plants had red flowers? Is this an unusual occurrence?
e. If you actually observed 53 of 100 offspring plants with red flowers, and if you were certain that the genetic ratio 3: 1 was correct, what other explanation could you give for this unusual occurrence?

A purchaser of electric relays buys from two suppliers, $A$ and $B$. Supplier $A$ supplies two of every three relays used by the company. If 75 relays are selected at random from those in use by the company, find the probability that at most 48 of these relays come from supplier A. Assume that the company uses a large number of relays.

Is television dangerous to your diet? Psychologists believe that excessive eating may be associated with emotional states (being upset or bored) and environmental cues (watching television, reading, and so on). To test this theory, suppose you randomly selected 60 overweight persons and matched them by weight and gender in pairs. For a period of 2 weeks, one of each pair is required to spend evenings reading novels of interest to him or her. The other member of each pair spends each evening watching television. The calorie count for all snack and drink intake for the evenings is recorded for each person, and you record $x=19,$ the number of pairs for which the television watchers' calorie intake exceeded the intake of the readers. If there is no difference in the effects of television and reading on calorie intake, the probability $p$ that the calorie intake of one member of a pair exceeds that of the other member is . $5 .$ Do these data provide sufficient evidence to indicate a difference between the effects of television watching and reading on calorie intake? (HINT: Calculate the $z$ -score for the observed value, $x=19 .)$

The Biology Data Book reports that the gestation time for human babies averages 278 days with a standard deviation of 12 days. $^{8}$ Suppose that these gestation times are normally distributed.
a. Find the upper and lower quartiles for the gestation times.
b. Would it be unusual to deliver a baby after only 6 months of gestation? Explain.

In Exercise 6.28 , we suggested that the IRS assign auditing rates per state by randomly selecting 50 auditing percentages from a normal distribution with a mean equal to $1.55 \%$ and a standard deviation of $.45 \%$
a. What is the probability that a particular state would have more than $2 \%$ of its tax returns audited?
b. What is the expected value of $x$, the number of states that will have more than $2 \%$ of their income tax returns audited?
c. Is it likely that as many as 15 of the 50 states will have more than $2 \%$ of their income tax returns audited?

There is a difference in sports preferences between men and women, according to a recent survey. Among the 10 most popular sports, men include competition-type sports-pool and billiards, basketball, and softball-whereas women include aerobics, running, hiking, and calisthenics. However, the top recreational activity for men was still the relaxing sport of fishing, with $41 \%$ of those surveyed indicating that they had fished during the year. Suppose 180 randomly selected men are asked whether they had fished in the past year.
a. What is the probability that fewer than 50 had fished?
b. What is the probability that between 50 and 75 had fished?
c. If the 180 men selected for the interview were selected by the marketing department of a sporting goods company based on information obtained from their mailing lists, what would you conclude about the reliability of their survey results?

A psychological introvert-extrovert test produced scores that had a normal distribution with a mean and standard deviation of 75 and $12,$ respectively. If we wish to designate the highest $15 \%$ as extroverts, what would be the proper score to choose as the cutoff point?

Students very often ask their professors whether they will be "curving the grades." The traditional interpretation of "curving grades" required that the grades have a normal distribution, and that the grades will be assigned in these proportions:
$$
\begin{array}{l|lllll}
\text { Letter Grade } & \mathrm{A} & \mathrm{B} & \mathrm{C} & \mathrm{D} & \mathrm{F} \\
\hline \text { Proportion of Students } & 10 \% & 20 \% & 40 \% & 20 \% & 10 \%
\end{array}
$$
a. If the average "C" grade is centered at the average grade for all students, and if we assume that the grades are normally distributed, how many standard deviations on either side of the mean will constitute the "C" grades?
b. How many deviations on either side of the mean will be the cutoff points for the "B" and "D" grades?

Refer to Exercise $6.83 .$ For ease of calculation, round the number of standard deviations for " $\mathrm{C}$ " grades to ±.5 standard deviations, and for " $\mathrm{B}$ " and "D" grades to ±1.5 standard deviations. Suppose that the distribution of grades for a large class of students has an average of 78 with a standard deviation of 11 . Find the appropriate cutoff points for the grades $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D},$ and $\mathrm{F}$.

In Exercise $1.67,$ Allen Shoemaker derived a distribution of human body temperatures, which has a distinct mound-shape. Suppose we assume that the temperatures of healthy humans is approximately normal with a mean of 98.6 degrees and a standard deviation of 0.8 degrees.
a. If a healthy person is selected at random, what is the probability that the person has a temperature above 99.0 degrees?
b. What is the 95 th percentile for the body temperatures of healthy humans?

Calculate the area under the standard normal curve to the left of these values:
a. $z=-.90$
b. $z=2.34$
c. $z=5.4$

Calculate the area under the standard normal curve between these values:
a. $z=-2.0$ and $z=2.0$
b. $z=-2.3$ and -1.5

Find the following probabilities for the standard normal random variable $z$ :
a. $P(-1.96 \leq z \leq 1.96)$
b. $P(z>1.96)$
c. $P(z<-1.96)$

a. Find a $z_{0}$ such that $P\left(z>z_{0}\right)=.9750 .$
b. Find a $z_{0}$ such that $P\left(z>z_{0}\right)=.3594$.

a. Find a $z_{0}$ such that $P\left(-z_{0} \leq z \leq z_{0}\right)=.95$.
b. Find a $z_{0}$ such that $P\left(-z_{0} \leq z \leq z_{0}\right)=.98$.

A normal random variable $x$ has mean $\mu=5$ and $\sigma=2$. Find the following probabilities of these $x$ -values:
a. $1.2<x<10$
b. $x>7.5$
c. $x \leq 0$

Let $x$ be a binomial random variable with $n=$ 36 and $p=.54 .$ Use the normal approximation to find:
a. $P(x \leq 25)$
b. $P(15 \leq x \leq 20)$
c. $P(x>30)$

A Snapshot in $U S A$ Today indicates that $51 \%$ of Americans say the average person is not very considerate of others when talking on a cellphone. ${ }^{10}$ Suppose that 100 Americans are randomly selected.
a. Use the Calculating Binomial Probabilities applet from Chapter 5 to find the exact probability that 60 or more Americans would indicate that the average person is not very considerate of others when talking on a cellphone.
b. Use the Normal Approximation to Binomial Probabilities to approximate the probability in part a. Compare your answers.

Philatelists (stamp collectors) often buy stamps at or near retail prices, but, when they sell, the price is considerably lower. For example, it may be reasonable to assume that (depending on the mix of a collection, condition, demand, economic conditions, etc.) a collection will sell at $x \%$ of the retail price, where $x$ is normally distributed with a mean equal to $45 \%$ and a standard deviation of $4.5 \% .$ If a philatelist has a collection to sell that has a retail value of $\$ 30,000,$ what is the probability that the philatelist receives these amounts for the collection?
a. More than $\$ 15,000$
b. Less than $\$ 15,000$
c. Less than $\$ 12,000$

The scores on a national achievement test were approximately normally distributed, with a mean of 540 and a standard deviation of $110 .$
a. If you achieved a score of 680 , how far, in standard deviations, did your score depart from the mean?
b. What percentage of those who took the examination scored higher than you?

Although faculty salaries at colleges and universities in the United States continue to rise, they do not always keep pace with the cost of living nor with salaries in the private sector. In 2005 , the National Center for Educational Statistics indicated that the average salary for Assistant Professors at public four-year colleges was $\$ 50,581 .^{11}$ Suppose that these salaries are normally distributed with a standard deviation of $\$ 4000 .$
a. What proportion of assistant professors at public 4-year colleges will have salaries less than $\$ 45,000 ?$
b. What proportion of these professors will have salaries between $\$ 45,000$ and $\$ 55,000 ?$

Briggs and King developed the technique of nuclear transplantation, in which the nucleus of a cell from one of the later stages of the development of an embryo is transplanted into a zygote (a single-cell fertilized egg) to see whether the nucleus can support normal development. If the probability that a single transplant from the early gastrula stage will be successful is .65, what is the probability that more than 70 transplants out of 100 will be successful?