Weiers TB Ch7

Chapter 7: Continuous Probability Distributions

TRUE/FALSE

1.True or FalseIn any normal distribution, the mean, median, mode, and standard deviation are all at the same position on the horizontal axis.

ANS:FPTS:1OBJ:Section 7.2

2.True or FalseIn the normal distribution, the curve is asymptotic but never intercepts the horizontal axis either to the left or right.

ANS:TPTS:1OBJ:Section 7.2

3.True or FalseIn the normal distribution, the total area beneath the curve represents the probability for all possible outcomes for a given event.


4.True or FalseUnder certain conditions, the normal distribution can be used to approximate the binomial distribution.


5.True or FalseThere exists a different normal curve for every possible pair of and .


6.True or FalseThe standard deviation of the binomial distribution with n = 25 and = .50 is 6.25.


7.True or FalseThe mean of the exponential distribution is the inverse of that of the normal distribution.


8.True or FalseIf Excel and Minitab packages are both used to generate its own sample of 10,000 observations from a theoretical continuous distribution, the two sets of observations are expected to be identical.


9.True or FalseAccording to the law of large numbers, frequency distributions generated by statistical packages will more closely approach the theoretical distribution from which they were drawn whenever the number of observations becomes larger.


10.True or FalseContinuous probability distributions describe probabilities associated with random variables that can take on any value along a given range or continuum and for which there are no gaps between these possible values.


11.True or FalseFor any specified interval of values, the probability that a continuous random variable x will assume a value within the interval is the area beneath the curve between the two points describing the interval.


12.True or FalseAny normal distribution is a symmetrical, bell-shaped curve, with mean = 0.0 and standard deviation = 1.0.


13.True or FalseRegardless of the shape of a particular normal curve, about 75% of the area is within the interval , where and are the mean and the standard deviation, respectively.


14.True or FalseThe standard normal distribution has a mean of 0.0 and standard deviation of 1.0.


15.True or FalseThe exponential distribution is a family of discrete distributions, all of which are negatively skewed.


16.True or FalseComputer statistical packages, such as Excel and Minitab, can determine the exact area beneath a specified portion of the continuous distribution curve, or probability density function, and eliminate the need for calculations or table references.


MULTIPLE CHOICE

1.A continuous probability distribution represents a random variable:a.having outcomes, which occur in counting numbers.

b.having an infinite number of outcomes which may assume any number of values within an interval.

c.which is best described in a histogram.

d.which has a definite probability for the occurrence of a given integer.

e.None of these is correct.

ANS:BPTS:1OBJ:Section 7.1

2.Which of the following are not correct concerning the probability distribution for any continuous random variable?a.The vertical coordinate is the probability density function.

b.The range of the random variable is found on the y-axis.

c.The total area represented under the curve will be equal to 1.00.

d.The probability that x will take on a value between a and b will be the area under the curve between points a and b.

e.The area under the curve represents the sum of probabilities for all possible outcomes.


3.In a continuous probability distribution, the probability that x will take on an exact value:a.is equal to the height of the curve at that value.

b.is calculated using the probability density.

c.is always equal to 0.

d.is always greater than 0.

e.None of these is correct.

ANS:CPTS:1OBJ:Section 7.1

4.Which of the following statements are true regarding the areas beneath any normal curve?a.About 68.3% of the area is in the interval to .

b.About 95.5% of the area is in the interval to .

c.Nearly all of the area (about 99.7%) is in the interval to

d.All of these are true.

e.None of these is true.

ANS:DPTS:1OBJ:Section 7.2

5.Which of the following is not a characteristic for a normal distribution?a.It is symmetrical distribution

b.The mean is always zero

c.The mean, median, and mode are all equal

d.It is a bell-shaped distribution

e.The area under the curve always equals 1.0


6.A videotape store has an average weekly gross of $1,158 with a standard deviation of $120. Let x be the store's gross during a randomly selected week. If this is a normally distributed random variable, then the number of standard deviations from $1,158 to $1,360 is:a.0.4535.

b.0.0465.

c.20.98.

d.1.683.

e.none of these.


7.Using the standard normal table, the total area between z = -0.75 and z = 1.21 is:a.0.2734

b.0.3869

c.0.3397

d.0.2266

e.0.6603

ANS:EPTS:1OBJ:Section 7.3

8.Let z1 be a z score that is unknown but identifiable by position and area. If the area to the right of z1 is 0.8413, then the value of z1 must be:a.1.00.

b.-1.00.

c.0.00.

d.0.41.

e.-0.41.


9.Using the standard normal table, the total area between z = -1.82 and z = -1.38 is:a.0.0494

b.0.4656

c.0.1554

d.0.4162

e.0.1005

ANS:APTS:1OBJ:Section 7.3

10.If the area to the right of a positive z1 is 0.0869, then the value of z1 must be:a.-1.36.

b.1.36.

c.1.71.

d.1.80.

e.0.22.


11.The z-score representing the 10th percentile of the standard normal distribution is:a.0.255.

b.-0.255.

c.-1.645.

d.1.28.

e.-1.28.



b.-0.67.

c.1.28.

d.-1.28.

e.1.645.



b.-1.28.

c.0.67.

d.-0.67.

e.1.645.


14.For a normal curve, if the mean is 20 minutes and the standard deviation is 5 minutes, then the area between 22 and 25 minutes is:a.0.3413.

b.0.1554.

c.0.4967.

d.0.1859.

e.0.2248.


15.For a normal curve, if the mean is 20 minutes and the standard deviation is 5 minutes, then the area between 11 and 19 minutes is:a.0.4641.

b.0.0359.

c.0.3848.

d.0.8848.

e.0.9641.


16.A bakery firm finds that its average weight of the most popular package of cookies is 32.06 ounces with a standard deviation of 0.1 ounces. What portion of cookie packages will weigh less than 31.9 ounces or more than 32.3 ounces?a.0.8000

b.0.4452

c.0.4918

d.0.9370

e.0.0630


17.A retailer finds that the demand for a very popular board game averages 100 per week with a standard deviation of 20. If the seller wishes to have adequate stock 95% of the time, how many of the games must she keep on hand?a.139.2

b.100.0

c.132.9

d.195.0

e.125.6


18.A manufacturer of tow chains finds that the average breaking point is at 3,500 pounds and the standard deviation is 250 pounds. If you pull a weight of at least 4200 pounds with this tow chain, what percentage of the time would you expect the chain to break?a.2.8%

b.0.26%

c.49.74%

d.99.74%

e.50%


19.The average labor time to sew a pair of denim jeans is 4.2 hours with a standard deviation of 30 minutes. If the distribution is normal, then the probability of a worker finishing a pair of jeans in less than 3.5 hours is:a.0.0808.

b.0.4192.

c.0.5808.

d.0.9192.

e.0.9808.


20.A salesman who uses his car extensively finds that his gasoline bills average $125.32 per month with a standard deviation of $49.51. The probability that his bill will be less than $50 a month or more than $150 for a single month is:a.0.4357.

b.0.1915.

c.0.6272.

d.0.3728.

e.0.6915.


21.Given that z is a standard normal random variable and that the area to the right of z is 0.1949, then the value of z must be:a.0.51.

b.-0.51.

c.0.86.

d.-0.86.


22.Using the standard normal table, the total area between z = 0.45 and z = 1.05 is:a.0.1736

b.0.3531

c.0.3264

d.0.6469

e.0.1795


23.Given that z is a standard normal random variable, a negative value of z indicates that:a.The value z is to the left of the mean.

b.The value z is to the right of the median.

c.The standard deviation of z is negative.

d.The area between zero and z is negative.

e.The probability associated with z is negative.


24.Given that z is a standard normal random variable, and that the area to the right of z is 0.9066, then the value of z must be:a.1.32.

b.-1.32.

c.0.66.

d.-0.66.


25.The proportion of the data from a standard normal distribution that falls within two standard deviations from the mean is:a.0.3413.

b.0.4772.

c.0.6826.

d.0.9544


26.If the z-value for a given value x of the random variable x is z = 1.96, and the distribution of x is normally distributed with a mean of 60 and a standard deviation of 6, then the x-value that this z-value corresponds to is:a.71.76.

b.67.96.

c.61.96.

d.48.24.


27.Your high school graduating class had 564 members. Thirty-three percent of these are expected to attend college. If the number of students who attend college follows the normal distribution, the probability that less than 160 will attend college is:a.0.3610.

b.0.4900.

c.0.0090.

d.0.0832.

e.0.2500


28.A large mail house which mails such items as catalogues, magazines, and other bulk mailings guarantees that there will be no more than a 3% error rate on its mailing labels. A customer who contracted a mailing to 190,000 individuals experienced a return of 5,900 items, which had incorrect addresses. Which of the following statements is true?a.There is a 3% chance of an incorrect return.

b.There is a 0.0054 probability that a return of 5900 or more incorrect addresses could occur if the true error rate is 3%.

c.There is a 0.4964 probability that a return of 5900 or more incorrect addresses could occur if the true error rate is 3%.

d.There is a 2.69 percent possibility that a return of 5900 incorrect addresses could occur if the true error rate is 3%.

e.None of these statements are correct.


29.In your college campus, 30% of the students use tobacco in some form. In your statistics class of 40 students, the probability of finding at least 15 students who use tobacco is:a.0.3051.

b.0.1949.

c.0.8051.

d.0.1515.

e.0.3485.


30.A zipper manufacturer has found that 1.5% of its zippers are defective. The company averages 4982 zippers per day. The probability of finding between 90 and 100 defects in a given day is:a.0.4981.

b.0.0310.

c.0.0360.

d.0.5310.

e.0.4671.


31.Which of the following is not a correct statement?a.The exponential distribution describes the Poisson process as a continuous random variable.

b.The exponential distribution is a family of curves, which are completely described by the mean.

c.The mean of the exponential distribution is the complement of the mean of the Poisson.

d.The Poisson is a probability distribution or a discrete random variable while the exponential distribution is continuous.

e.All of these are correct.


32.A local radio station maintains a very popular phone service, which provides callers with current weather reports. The incoming calls have a Poisson distribution with an average of 5 calls for each 15 minute-period. If x = time between calls, then the probability of receiving four or fewer calls in the next 15 minutes is:a.0.2341.

b.0.7659.

c.0.2636.

d.0.7364.

e.0.9502.


33.A dispatcher for an airport shuttle averages sending a van to the airport 2 times every 60 minutes. The distribution is Poisson, and the driver must take a 15-minute lunch break. The probability that he can finish lunch before getting a call is:a.0.1350.

b.0.6065.

c.0.3935.

d.1.6490.

e.0.8650.


34.A very large logging operation has serious problems keeping their skidders operating properly. The equipment fails at the rate of 3 breakdowns every 48 hours. Assume that x is time between breakdowns and is exponentially distributed. The probability of two or less breakdowns in the next 48-hour period is:a.0.9672.

b.0.0307.

c.0.2231.

d.0.7769.


35.If the mean of an exponential distribution is 2, then the value of the parameter is:a.4.0.

b.2.2.

c.1.0.

d.0.5.


36.If the random variable x is exponentially distributed with parameter = 4, then P(x 0.25), up to 4 decimal places, is:a.0.6321.

b.0.3679.

c.0.2500.

d.0.5000.


37.If the random variable x is exponentially distributed with parameter = 1.5, then P(2 x 4), up to 4 decimal places, is:a.0.6667.

b.0.0473.

c.0.5000.

d.0.2500.


38.Which of the following is not true for an exponential distribution with parameter ?a.Mean = .

b.Standard deviation = .

c.The distribution is completely determined once the value of is known.

d.The distribution is a two-parameter distribution since the mean and standard deviation are equal.


39.Like the normal distribution, the exponential density function f(x):a.is bell-shaped.

b.is symmetrical.

c.approaches infinity as x approaches zero.

d.approaches zero as x approaches infinity.


40.Which of the following distributions is suitable to model the length of time that elapses before the first telephone call is received by a switchboard?a.Exponential

b.Normal

c.Poisson

d.Binomial

e.Hypergeometric


41.The mean of the exponential distribution equals the mean of the Poisson distribution only when the former distribution has a mean equals:a.1.00.

b.0.50.

c.0.25.

d.2.00.


42.Which of the following distributions is appropriate to measure the length of time between arrivals at a grocery checkout counter?a.Binomial

b.Normal

c.Exponential

d.Poisson

e.Hypergeometric


43.Which of the following statements are true regarding the normal distribution curve?a.It is symmetrical.

b.It is bell-shaped.

c.It is asymptotic in that each end approaches the horizontal axis, but never reaches it.

d.Its mean, median, and mode are located at the same position.

e.All of these statements are true.


NUMERIC RESPONSE

1.Assume x is normally distributed with mean = 15 and standard deviation = 3. Use the approximate areas beneath the normal curve, as discussed in this section, to find P(x 15).

ANS:0.50

PTS:1OBJ:Section 7.2

2.Assume x is normally distributed with mean = 15 and standard deviation = 3. Use the approximate areas beneath the normal curve, as discussed in this section, to find P(12 x 18).

ANS:0.683



ANS:0.0225


4.Assume x is normally distributed with mean = 15 and standard deviation = 3. Use the approximate areas beneath the normal curve, as discussed in this section, to find P(x = 20).

ANS:0


5.Assume x is normally distributed with mean = 15 and standard deviation = 3. Use the approximate areas beneath the normal curve, as discussed in this section, to find P(9 x 21).

ANS:0.955



ANS:0.8415


7.In 2000, the average charge of tax preparation was $95. Assuming a normal distribution and a standard deviation of $10, use the approximate areas beneath the normal curve, as discussed in this section, to answer: What proportion of tax preparation fees were more than $95?

ANS:0.50


8.In 2000, the average charge of tax preparation was $95. Assuming a normal distribution and a standard deviation of $10, use the approximate areas beneath the normal curve, as discussed in this section, to answer: What proportion of tax preparation fees were between $75 and $115?

ANS:0.955



ANS:0.683


10.In 2000, the average charge of tax preparation was $95. Assuming a normal distribution and a standard deviation of $10, use the approximate areas beneath the normal curve, as discussed in this section, to answer: What proportion of tax preparation fees were more than $115?

ANS:0.0225



ANS:0.2417


12.Using the standard normal curve, find the area to the left of z = -2.26.

ANS:0.0119


13.Using the standard normal curve, find the area between z = 0 and z = 3.3.

____________________.

ANS:0.50


14.Determine P(z 1.58) for the standard normal curve.

ANS:0.0571


15.For the standard normal curve, determine the area to the right of z = 0.34.

ANS:0.3669


16.Assuming a standard normal distribution, determine P(z -0.75).

ANS:0.2266


17.Determine P(-2.45 z 1.69), assuming a standard normal distribution.

ANS:0.9474


18.Determine P(-2.45 z 0), assuming a standard normal distribution.

ANS:0.4929


19.Assume the standard normal curve and determine the probability or area between z = -1.28 and z = 1.28.

ANS:0.7994


20.Find z1 if the area to the left of a positive z1 is 0.9131.

ANS:1.36


21.Find the z-score that determines that the area to the left of z is 0.9292.

ANS:1.47


22.Find the z-score that determines that the area to the right of z is 0.8264.

ANS:-0.94


23.A continuous random variable x is normally distributed with a mean of 120 grams and a standard deviation of 30 grams. Convert x = 106 into its corresponding z-score.

ANS:-0.47


24.A continuous random variable x is normally distributed with a mean of $250 and a standard deviation of $50. Find the z-score for x = 175.

ANS:-1.50


25.A continuous random variable x is normally distributed with a mean of $250 and a standard deviation of $50. Find the z-score for x = 285.

ANS:0.70


26.For a normal curve, if the mean is 20 minutes and the standard deviation is 5 minutes, find the area to the left of 10 minutes.

ANS:0.0228


27.For a normal curve, if the mean is 20 minutes and the standard deviation is 5 minutes, find the area to the right of 12 minutes.

ANS:0.9452


28.The age of customers for a particular retail store follows a normal distribution with a mean of 37.5 years and a standard deviation of 7.6 years. What is the probability that the next customer who enters the store will be more than 31 years old?

ANS:0.8051


29.The age of customers for a particular retail store follows a normal distribution with a mean of 37.5 years and a standard deviation of 7.6 years. What is the probability that the next customer who enters the store will be less than 42 years old?

ANS:0.7224


30.The age of customers for a particular retail store follows a normal distribution with a mean of 37.5 years and a standard deviation of 7.6 years. What is the probability that the next customer who enters the store will be between 40 and 45 years old?

ANS:0.2096


31.The average waiting time at the checkout counter for a large grocery chain is 2.45 minutes with a standard deviation of 24 seconds (0.40 minutes). Assume that the distribution of waiting time is normal. What is the probability that a customer must wait more than 3 minutes for check out?

ANS:0.0838


32.The average waiting time at the checkout counter for a large grocery chain is 2.45 minutes with a standard deviation of 24 seconds (0.40 minutes). Assume that the distribution of waiting time is normal. What proportion of the customers are served in between 1 minute and 2.5 minutes?

ANS:0.5517


33.Suppose the monthly demand for automobile tires at a tire dealer is normally distributed with a mean of 250 tires and a standard deviation of 50 tires. How many tires must the store have in inventory at the beginning of each month in order to meet demand 95 percent of the time?

ANS:332


34.A company sells toothpaste in a tube advertised to contain 8 ounces. The tube filling process is set with a mean of 8.21 ounces. In this continuous production process, the amount of toothpaste put in a tube is normally distributed with a mean of 8.21 ounces and a standard deviation of 0.09 ounces. If the actual capacity of the tubes used is 8.45 ounces, what proportion of the tubes will be filled beyond capacity?

ANS:0.0038


35.A manufacturer of nails packages 16-penny nails in 25-pound boxes. Filling of the boxes is automated with the dispenser set to an average of 25.1 pounds with a standard deviation of 0.25 pounds. One 25-pound box of 16-penny nails is included in the estimate for a remodeling contract by a local builder. What is the probability that the box purchased will have less than the required 25 pounds?

ANS:0.3446


36.The Independent Bank surveyed the status of student accounts and found that the average overdraft was $21.22 with a standard deviation of $5.49. If the distribution is normal, find the probability of a student being overdrawn by more than $18.75.

ANS:0.6736


37.A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 175 feet and a standard deviation of 15 feet. His landing net is 50 feet long and the mid-point of the net is positioned 175 feet from the cannon. What is the probability that the performer will hit the net on a given night?

ANS:0.9050


38.A circus performer who gets shot from a cannon is supposed to land in a safety net positioned at the other end of the arena. The distance he travels is normally distributed with a mean of 175 feet and a standard deviation of 15 feet. His landing net is 50 feet long and the mid-point of the net is positioned 175 feet from the cannon. What is the probability that the performer will miss the net on a given night?

ANS:0.0950


39.The recent average starting salary for new college graduates in computer information systems is $47,500. Assume salaries are normally distributed with a standard deviation of $4,500. What is the probability of a new graduate receiving a salary between $45,000 and $50,000?

ANS:0.4246


40.The speed of cars passing through a checkpoint follows a normal distribution with a mean of 62.6 miles per hour and a standard deviation of 3.7 miles per hour. What is the probability that the next car passing by will be exceeding 65.5 miles per hour?

ANS:0.2177


41.The speed of cars passing through a checkpoint follows a normal distribution with a mean of 62.6 miles per hour and a standard deviation of 3.7 miles per hour. What is the probability that the next car passing by will be exceeding 58.1 miles per hour?

ANS:0.8880


42.The speed of cars passing through a checkpoint follows a normal distribution with a mean of 62.6 miles per hour and a standard deviation of 3.7 miles per hour. What is the probability that the next car passing by will be traveling between 61 and 70 miles per hour?

ANS:0.6436


43.The recent average starting salary for new college graduates in computer information systems is $47,500. Assume salaries are normally distributed with a standard deviation of $4,500. What is the probability of a new graduate receiving a salary between $45,000 and $50,000?

ANS:0.4246


44.A manufacturer of washing machines has experienced a 2% repair rate. In a city where 120,000 of its machines are located, the company plans to open its own repair service and needs to determine the number of repair workers to hire. Each worker can handle calls for 1000 machine repairs. If the company wants to cover calls 90% of the time, how many workers must it hire for the new repair facility?

ANS:2.46


45.An elementary school teacher learned that 40 percent of school age children have at least 3 cavities. The teacher has 30 students in his class. How many students would he expect in his class to have at least three cavities?

ANS:12


46.An elementary school teacher learned that 40 percent of school age children have at least 3 cavities. The teacher has 30 students in his class. What is the standard deviation for the number of school age children that have at least 3 cavities?

ANS:2.683


47.The selling price of various homes in a community follows the normal distribution with a mean of $176,000 and a standard deviation of $22,300. What is the probability that the next house will sell for less than $190,000?

ANS:0.7357


48.The selling price of various homes in a community follows the normal distribution with a mean of $176,000 and a standard deviation of $22,300. What is the probability that the next house will sell for less than $158,000?

ANS:0.2090


49.The selling price of various homes in a community follows the normal distribution with a mean of $176,000 and a standard deviation of $22,300. What is the probability that the next house will sell for between $150,000 and $168,000?

ANS:0.2384


50.An elementary school teacher learned that 40 percent of school age children have at least 3 cavities. The teacher has 30 students in his class. What is the probability that 10 students from this class have at least 3 cavities?

ANS:0.1115


51.Suppose that x is a binomial random variable with n = 100 and = 0.18. Employ the normal approximation to find P(15 x 21).

ANS:0.6976


52.Let x be a binomial random variable. Find P(x 40) if n = 100 and = 0.32.

ANS:0.0537


53.Women make up 58% of the U. S. civilian workforce of 124 million. The U. S. Department of Commerce randomly selects 100 workers for a conference on national health care. What is the probability that more than 20 of these workers are female?

ANS:0.9999 or approximately 1


54.Women make up 58% of the U. S. civilian workforce of 124 million. The U. S. Department of Commerce randomly selects 100 workers for a conference on national health care. If there are fewer than 45 or more than 65 females invited, severe political ramifications are possible. What is the probability that the conference planners will be criticized for the representation by women at the conference?

ANS:0.0674


55.A coin is flipped 14 times. Using the normal distribution, calculate the probability of observing either 4, 5, or 6 heads.

ANS:0.3629


56.An efficiency expert makes periodic checks for weighting errors for a long distance shipping firm. The expert inspects for errors in weighing, in recording weights, and errors in processing the bills of lading. Based on past records, the number of weekly errors for all shipments averages 5.30 with a standard deviation of 1.23, and the frequency histogram approximates a normal distribution. Suppose x is the number of weighing errors that will occur next week. Compute the approximate probability for (x = 6).

ANS:0.2729


57.A statistics class is composed of 60 percent females. If 15 students are selected randomly, what is the probability that this group will include either 8, 9, 10, or 11 females? Use the normal distribution.

ANS:0.6916


58.An efficiency expert makes periodic checks for weighting errors for a long distance shipping firm. The expert inspects for errors in weighing, in recording weights, and errors in processing the bills of lading. Based on past records, the number of weekly errors for all shipments averages 5.30 with a standard deviation of 1.23, and the frequency histogram approximates a normal distribution. Suppose x is the number of weighing errors that will occur next week.

Compute the approximate probability for (4 x 7).

ANS:0.8912


59.An efficiency expert makes periodic checks for weighting errors for a long distance shipping firm. The expert inspects for errors in weighing, in recording weights, and errors in processing the bills of lading. Based on past records, the number of weekly errors for all shipments averages 5.30 with a standard deviation of 1.23, and the frequency histogram approximates a normal distribution. Suppose x is the number of weighing errors that will occur next week.

Compute the approximate probability for (x 2).

ANS:0.0113


60.In a paper mill, which produces newsprint from Southern Pine, the production manager finds a flaw for each 1000 feet of newsprint. He knows that the outcome is a Poisson process with a continuous distribution. If x = feet between flaws, what is the probability of finding 600 or more feet between the next two flaws?

ANS:0.5488


61.In a paper mill, which produces newsprint from Southern Pine, the production manager finds a flaw for each 1000 feet of newsprint. He knows that the outcome is a Poisson process with a continuous distribution. A group of visitors is touring the plant. After arriving at the final stage of production, if they remain long enough to observe 330 feet of newsprint produced, what is the probability that they will miss seeing the next defect?

ANS:0.2811


62.In 1997, private planes had 1.5 fatal crashes per 100 thousand flying hours. For a continuous random variable with an exponential distribution: What is the probability that the time between the next two crashes will fall between 50 and 70 thousands hours, i.e. P(50 x 70); x = thousands of flying hours between fatal crashes.

ANS:0.1224


63.In 1997, private planes had 1.5 fatal crashes per 100 thousand flying hours. Assume a continuous random variable with a exponential distribution. A national association of private plane owners has 560 members who fly 1.5 million miles each year. The mileage crash rate has been estimated to be 20 per 100 million miles of air travel. What is the probability that the next crash by a member of the association will not occur until more than one year from now?

ANS:0.2592


64.Grades for a statistics exam for a certain class follow the normal probability distribution with a mean of 82 and a standard deviation of 12. What percentage of the students in the class had a grade 81 or more?

ANS:0.5319


65.Grades for a statistics exam for a certain class follow the normal probability distribution with a mean of 82 and a standard deviation of 12. What percentage of the students in the class earned a C grade (70-79)?

ANS:0.2426


66.Grades for a statistics exam for a certain class follow the normal probability distribution with a mean of 82 and a standard deviation of 12. What percentage of the students in the class earned a B grade (80-89)?

ANS:0.2865


67.The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes. What is the probability that it will take a technician less than 10 minutes to fix a computer problem?

ANS:0.4866


68.The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes. What is the probability that it will take a technician between 10 to 15 minutes to fix a computer problem?

ANS:0.1455


69.A manufacturer produces Walkman tap players/radios. The electronic components are positioned in the outer case after the case has been given an epoxy finish. The production manager must time the drying process with the total assembly line and therefore must determine the appropriate drying time. An intensive cost analysis indicated that a risk of just 0.3% of the cases being wet and therefore smudged was acceptable. If drying times are normally distributed with a mean of 3.6 minutes and a standard deviation of 0.6 minutes, what time setting should the production manager set for the automatic timer which controls the production line?

____________________ minutes

ANS:5.25


70.The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes. What is the variance of the time it takes a technician to fix a computer problem?

____________________ minutes

ANS:225


71.The total area beneath the curve of any continuous distribution is ____________________.

ANS:1.0


72.The probability that any continuous random variable x will take on any specific value along an interval is ____________________.

ANS:0.0


COMPLETION

1.In the normal distribution, the total area under the curve is equal to ____________________.

ANS:one1


2.In the normal distribution, the right half of the curve is a mirror image of the _________________________, since the distribution is ____________________.

ANS:left half; symmetric


3.If we ____________________ the normal curve, we express the original x values in terms of their number of standard deviations away from the mean.

ANS:standardize


4.The binomial distribution is symmetrical whenever the population proportion is ____________________, and approaches symmetry for values that are close to ____________________.

ANS:0.5; 0.5


5.For a Poisson process, the ____________________ distribution describes the continuous random variable x, where x is the amount of time, space, or distance between occurrences of these rare events.

ANS:exponential


SHORT ANSWER

1.In 2000, the average charge of tax preparation was $95. Assuming a normal distribution and a standard deviation of $10, use the approximate areas beneath the normal curve, as discussed in this section, to answer: What proportion of tax preparation fees were exactly $100?

ANS:P(x = 100) = 0


2.If z is a standard normal random variable, find the value z1 for which:

A) P(0 zz1) = 0 .276

B) P(zz1) = 0.341

C) P(zz1) = 0.819

D) P(-z1 Zz1) = 0.785

ANS:0.76; 0.41; -0.91; 1.24


3.If x is a normal random variable with a mean of 100 and a standard deviation of 10, find the following probabilities:

A) P(x 128)

B) P(x 113)

C) P(87 x 98)

ANS:0.0026; 0.9032; 0.3239


4.A computer statistical package has simulated 1000 random observations from a normal distribution with mean = 50 and standard deviation = 10. Sketch the approximate box-and-whisker plot for the resulting data.

ANS:


5.If a computer statistical package were to simulate 500 random observations from a normal distribution with mean = 100 and standard deviation = 50, what percentage of these observations would you expect to have a value of 200 or more? Do you think the actual number in the " 200" range would equal the expected number in this range? If so, why? If not, why not?

ANS:2.28%; No. We would expect 11.4 of the 500 observations (2.28%) to have a value of 200 or more. The actual number would not be equal to the expected number. However, the more observations we select, the closer we will tend to come to what we expect.


6.Scores of high school students on a national mathematics exam in Egypt were normally distributed with a mean of 86 and a standard deviation of 4.

A) What is the probability that a randomly selected student will have a score of 80 or higher?B) If there were 97,680 students with scores higher than 91, how many students took the test?

____________________ (Remove all commas from your answer before submitting).

ANS:0.9332; 925000


7.The time it takes a technician to fix a computer problem is exponentially distributed with a mean of 15 minutes. What is the probability density function for the time it takes a technician to fix a computer problem?

ANS:f(x) = (1/15) e-x/15, x 0


8.What are the mean and standard deviation of a normally distributed random variable, which has been "standardized"?

Mean = ____________________

SD = ____________________

ANS:0.0; 1


9.A certain brand of flood lamps has a lifetime that is normally distributed with a mean of 3,750 hours and a standard deviation of 300 hours.

A). What proportion of these lamps will last for more than 4,000 hours?

B).What lifetime should the manufacturer advertise for these lamps in order that only 2% of the lamps will burn out before the advertised lifetime?

____________________ hours (Remove all commas from your answer before submitting).

ANS:0.2033; 3132


10.The normal distribution is a very good approximation to the binomial distribution whenever ____________________ and ____________________ are 5.

ANS:n; n(1-)


11.Complete the following table indicating which procedure to use in calculating binomial probabilities:

nProcedure

Large0.2

Large0.0

Large0.9

Small0.0

ANS:

nProcedure

Large0.2 Use the normal approximation

Large0.0 Use the Poisson approximation

Large0.9 Use the Poisson approximation

Small0.0 Use the binomial probability formula


12.Although the binomial distribution is discrete and the normal distribution is continuous, the normal distribution is a good approximation to the binomial whenever both ____________________ and ____________________ are 5, where n = number of trials, and = the probability of success in any given trial, are the parameters of the binomial distribution.

ANS:n; n(1-


13.Regardless of the shape of a particular normal curve, about what percentage of the area is within , respectively, where is the mean and is the standard deviation.

ANS:95.5% and 99.7% respectively


ESSAY

1.What are continuous probability distributions?

ANS:Continuous probability distributions describe probabilities associated with random variables that are able to assume any of an infinite number of values along an interval.


2.Why is the probability that a continuous random variable takes on any specific value equal to zero?

ANS:The probability that a continuous random variable takes on any specific value is equal to zero because there is an infinite number of possible values.


3.Explain why the total area beneath a probability density function is equal to 1.0.

ANS:The area beneath the probability density function represents the probability of the random variable, x, being between . Since x must be between (this is a certain event), the area must be equal to 1.0.


4.Identify the distribution shown in the following graph. Indicate the approximate areas that will lie beneath the curve for the intervals shown.

ANS:


5.What does it mean when we say a random variable is standardized?

ANS:A random variable x is standardized when each value of x has (the mean of x) subtracted from it, and the difference is divided by (the standard deviation of x.)


6.Explain what we mean by saying a random variable is approximately normally distributed.

ANS:If the probabilities for the outcomes of the random variable are approximately equal to the areas under the normal curve, its distribution is approximately normal.


7.Express in your own words the procedure for finding the area under the standard normal curve between z = 0 and z = 1.53.

ANS:Use the standard normal table in text. Trace down the left hand column to "1.5" and go across the row until reaching the column headed at the top with ".03". The value is 0.4370, which is the area between z = 0 and z = 1.53.


8.Describe the method of finding the area under the standard normal curve between z = 0 and z =- 1.33.

ANS:Recall the property of symmetry. The area to the left of 0 to - 1.33 is the same as for the area to the right of 0 to 1.33. Go down the left most column of the standard normal table to the row with "1.3" and right across the rows until in the column headed at the top with ".03". The value is 0.4082, which is the area between z = 0 and z = -1.33.


9.Use the properties of the standard normal curve to describe the method of finding the area to the right of z = 1.42.

ANS:Since the standard normal table in text gives areas between 0 and positive values of z, to find the area to the right of a z value requires the use of the symmetry property which places .5 of the whole area to the right of z = 0. Trace down the left most column to the value "1.4" and across the row to the column headed by the value ".02". The value is 0.4222, which is the area between z = 0 and z = 1.42. Subtract this area from 0.5 to obtain the desired area of 0.0778.


10.For a value of z = -1.59, explain the steps needed to find the area to the left of this value.

ANS:By symmetry the area to the right of 0 is the same as the area to the left of 0 for values of z. In finding the area to the left of a negative value of z use the symmetry property to reason that .5 of the area lies to the left of 0. Go down the left most column to the value "1.5" and across the row until reaching the column headed by "9". The value is 0.4441, which is the area between z = 0 and z = -1.59. Subtract this area from 0.5 to obtain the desired area of 0.0559.


11.Why is it important to use the correction for continuity when approximating binomial probabilities with the normal distribution?

ANS:Because the binomial distribution has gaps between possible values of x (since it is discrete), while the normal distribution is continuous, the normal approximation to the binomial involves a correction for continuity. The correction consists of expanding each possible value of the discrete random variable x by 0.5 in each direction.


12.A continuous random variable x has the following probability density function:

f(x) = 2e-2x , x 0

What is the distribution of the random variable x?

ANS:Exponential distribution with = 2


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Weiers TB Ch7