7.93 The print on the package of 100-watt General Electric soft-white lightbulbs claims that these bulbs have an average life of 750 hours. Assume that the lives of all such bulbs have a normal distribution with a mean of 750 hours and a standard deviation of 55 hours. Let be the mean life of a random sample of 25 such bulbs. Find the mean and standard deviation of and describe the shape of its sampling distribution.
7.94 The times spent waiting in line by all drivers to get their licenses renewed at the motor vehicle department office in a city have a distribution that is skewed to the right with a mean of 24 minutes
x x zpˆp sp^ where sp^ A pq n pˆ sp^ A pq n pˆ mp^p pˆ
xnumber of elements in the sample that possess a specific characteristic ntotal number of elements in the sample
pˆ x n
Xnumber of elements in the population that possess a specific characteristic Ntotal number of elements in the population
p X N zxm sx where sx s 1n x sx s 1n x mxm x
and a standard deviation of 7 minutes. Let be the mean waiting time for a random sample of 100 drivers renewing their licenses at this office. Calculate the mean and standard deviation of and comment on the shape of its sampling distribution.
7.95 Refer to Exercise 7.93. The print on the package of 100-watt General Electric soft-white light- bulbs says that these bulbs have an average life of 750 hours. Assume that the lives of all such bulbs have a normal distribution with a mean of 750 hours and a standard deviation of 55 hours. Find the probability that the mean life of a random sample of 25 such bulbs will be
a. greater than 735 hours
b. between 725 and 740 hours
c. within 15 hours of the population mean
d. less than the population mean by 20 hours or more
7.96 Refer to Exercise 7.94. The times spent waiting in line by all drivers to get their licenses renewed at the motor vehicle department office of a large city have a distribution that is skewed to the right with a mean of 24 minutes and a standard deviation of 7 minutes. Find the probability that the mean waiting time for a random sample of 100 drivers to get their licenses renewed is
a. less than 22 minutes
b. between 23 and 26 minutes
c. within 1 minute of the population mean
d. greater than the population mean by 2 minutes or more
7.97 According to a 2002 survey by America Online, mothers with children under 18 years of age spent an average of 16.87 hours per week online (USA TODAY, May 7, 2002). Assume that the mean time spent online by all current mothers with children under 18 years of age is 16.87 hours per week with a standard deviation of 5 hours per week. Find the probability that the mean time spent online per week by a random sample of 100 such mothers is
a. greater than 17 hours
b. between 16.5 and 17.5 hours
c. within .75 hour of the population mean
d. less than the population mean by .75 hour or more
7.98 A machine at Keats Corporation fills 64-ounce detergent jugs. The probability distribution of the amount of detergent in these jugs is normal with a mean of 64 ounces and a standard deviation of .4 ounce. The quality control inspector takes a sample of 16 jugs once a week and measures the amount of detergent in these jugs. If the mean of this sample is either less than 63.75 ounces or greater than 64.25 ounces, the inspector concludes that the machine needs an adjustment. What is the probability that based on a sample of 16 jugs the inspector will conclude that the machine needs an adjustment when actually it does not?
7.99 Suppose that 88% of the cases of car burglar alarms that go off are false. Let be the proportion of false alarms in a random sample of 80 cases of car burglar alarms that go off. Calculate the mean and standard deviation of and describe the shape of its sampling distribution.
7.100 Seventy percent of adults favor some kind of government control on the prices of medicines. Assume that this percentage is true for the current population of all adults. Let be the proportion of adults in a random sample of 400 who favor government control on the prices of medicines. Calculate the mean and standard deviation of and describe the shape of its sampling distribution.
7.101 Refer to Exercise 7.100. Seventy percent of adults favor some kind of government control on the prices of medicines. Assume that this percentage is true for the current population of all adults.
a. Find the probability that the proportion of adults in a random sample of 400 who favor some kind of government control on the prices of medicines is
i. less than .65 ii. between .73 and .76
b. What is the probability that the proportion of adults in a random sample of 400 who favor some kind of government control is within .06 of the population proportion?
c. What is the probability that the sample proportion is greater than the population proportion by .05 or more? pˆ pˆ pˆ pˆ x x
7.102 According to a survey by computing service firm Automatic Data Processing, 40% of job appli- cants misrepresent their education or employment history on their résumés (USA TODAY, June 19, 2002). Assume that 40% of all current job applicants have such inaccuracies on their résumés. Let be the proportion of a random sample of 120 job applicants whose résumés contain such misrepresentations.
a. Find the probability that this sample proportion is within .05 of the population proportion.
b. What is the probability that this sample proportion is not within .05 of the population proportion?
c. Find the probability that this sample proportion is greater than the population proportion by .08 or more.
d. What is the probability that this sample proportion is less than the population proportion by .06 or more?
ADVANCED EXERCISES
7.103 Let be the mean annual salary of Major League Baseball players for 2002. Assume that the standard deviation of the salaries of these players is $105,000. What is the probability that the 2002 mean salary of a random sample of 100 baseball players was within $10,000 of the population mean,
? Assume that nN .05.
7.104 The test scores for 300 students were entered into a computer, analyzed, and stored in a file. Unfortunately, someone accidentally erased a major portion of this file from the computer. The only information that is available is that 30% of the scores were below 65 and 15% of the scores were above 90. Assuming the scores are normally distributed, find their mean and standard deviation.
7.105 A chemist has a 10-gallon sample of river water taken just downstream from the outflow of a chemical plant. He is concerned about the concentration,c(in parts per million), of a certain toxic sub- stance in the water. He wants to take several measurements, find the mean concentration of the toxic substance for this sample, and have a 95% chance of being within .5 part per million of the true mean value of c. If the concentration of the toxic substance in all measurements is normally distributed with
.8 part per million, how many measurements are necessary to achieve this goal?
7.106 A television reporter is covering the election for mayor of a large city and will conduct an exit poll (interviews with voters immediately after they vote) to make an early prediction of the outcome. Assume that the eventual winner of the election will get 60% of the votes.
a. What is the probability that a prediction based on an exit poll of a random sample of 25 vot- ers will be correct? In other words, what is the probability that 13 or more of the 25 voters in the sample will have voted for the eventual winner?
b. How large a sample would the reporter have to take so that the probability of correctly pre- dicting the outcome would be .95 or more?
7.107 A city is planning to build a hydroelectric power plant. A local newspaper found that 53% of the voters in this city favor the construction of this plant. Assume that this result holds true for the pop- ulation of all voters in this city.
a. What is the probability that more than 50% of the voters in a random sample of 200 voters selected from this city will favor the construction of this plant?
b. A politician would like to take a random sample of voters in which more than 50% would favor the plant construction. How large a sample should be selected so that the politician is 95% sure of this outcome?
7.108 Refer to Exercise 6.91. Otto is trying out for the javelin throw to compete in the Olympics. The lengths of his javelin throws are normally distributed with a mean of 290 feet and a standard deviation of 10 feet. What is the probability that the total length of three of his throws will exceed 885 feet?
7.109 A certain elevator has a maximum legal carrying capacity of 6000 pounds. Suppose that the population of all people who ride this elevator have a mean weight of 160 pounds with a standard de- viation of 25 pounds. If 35 of these people board the elevator, what is the probability that their com- bined weight will exceed 6000 pounds? Assume that the 35 people constitute a random sample from the population.