ES9 Additional Exercises

Chapter 6

1. According to the November 1993 issue of Harper's magazine, our kids spend from 1200 to 1800 hours a year in front of the television set. Suppose the time spent by kids in front of the television set per year is normally distributed with a mean equal to 1500 hours and a standard deviation equal to 100 hours.

a. What percentage spend between 1400 and 1600 hours?

b. What percentage spend between 1300 and 1700 hours?

c. What percentage spend between 1200 and 1800 hours?

2. 
   Compare the results (a) through (c) with the empirical rule. Explain the relationship.
   

2. For a particular age group of adult males, the distribution of cholesterol readings, in mg/dl, is normally distributed with a mean of 210 and a standard deviation of 15.

a. What percentage of this population would have readings exceeding 250?

b. What percentage would have readings less than 150?

3. At Pacific Freight Lines, bonuses are given to billing clerks when they complete 300 or more freight bills during an eight-hour day. The number of bills completed per clerk per eight-hour day is approximately normally distributed with a mean of 270 and a standard deviation of 16. What proportion of the time should a randomly selected billing clerk expect to receive a bonus?

4. The waiting time x at a certain bank is approximately normally distributed with a mean of 3.7 min and a standard deviation of 1.4 min.

a. Find the probability that a randomly selected customer has to wait less than 2.0 min.

b. Find the probability that a randomly selected customer has to wait more than 6 min.

2. 
   Find the value of the 75th percentile for x.
   

5. According to a USA Snapshot ® (10-26-94), the average annual salary for a worker in the United States is $26,362. If we assume that the annual salaries for Americans are normally distributed with a standard deviation equal to $6,500, find the following:

a. What percentage earn below $15,000?

2. 
   What percentage earn above $40,000?
   

6. According to the 1991 issue of American Hospital Administration Hospital Statistics, the average daily census total for 116 hospitals in Mississippi equals 10,872. Suppose the standard deviation of the daily census totals for these hospitals equals 1505 patients. If the daily census totals are normally distributed:

a. What percentage of the days does the daily census total less than 8500 patients in these hospitals? Approximately how often should we expect this to occur?

b. What percentage of the days does the daily census total exceed 12,500 patients in these hospitals? Approximately how often should we expect this to occur?

8. A survey in the March 1994 issue of Life magazine indicated that 9 out of 10 Americans pray frequently and earnestly, and almost all say God has answered their prayers. Assuming "9 out of 10" is accurate, use the normal approximation to the binomial to find the probability that in a national survey of 1000 Americans, at least 925 will indicate that they pray frequently and earnestly.

9. An article in Life magazine indicated than 60% of Americans have had a psychic experience. An example of a psychic experience is dreaming about an event before it actually occurs. Some experts call psychic experiences precognitions, whereas others write it off as pure coincidence. Suppose a national survey of 2000 Americans is conducted and each is asked whether or not they have had a psychic experience. Use the normal approximation to the binomial distribution to find the probability that over 1,230 report such a phenomenon.

Source: Life, June, 1998. p. 89.

10. According to the Bureau of Justice Statistics Sourcebook of Criminal Justice Statistics 1992, 4.5% of young adults reported using alcohol daily for the past 30 days. Use the normal approximation to the binomial distribution to find the probability that, in a national poll of 1024 young adults, between 35 and 50 inclusive will indicate that they have used alcohol daily for the past 30 days.

a. Solve using normal approximation and Table 3.

b. Solve using a computer or calculator and the normal approximation method.

2. 
   Solve using a computer or calculator and the binomial probability function.
   

11. An article in USA Today (4-4-91) quoted a study involving 3365 people in Minneapolis-St. Paul between 1980 and 1982 and another 4545 between 1985 and 1987. It found that the average cholesterol level for males was 200. The authors of the study say the results of their study are probably similar nationwide. Assume that the cholesterol values for males in the United States are normally distributed with a mean equal to 200 and a standard deviation equal to 25.

a. What percentage have readings between 150 and 225?

b. What percentage have readings that exceed 250?

12. If 60% of the registered voters plan to vote for Ralph Brown for mayor of a large city. What is the probability that less than half of the voters, in a poll of 200 registered voters, plan to vote for Ralph brown?

Chapter 7

1. According to a USA Snapshot® (USA Today, October 21-23, 1994), the average amount spent per month for long-distance calls through the long-distance carrier is $31.65. If the standard deviation for long-distance calls through the long-distance carrier is $12.25 and a sample of 150 customers is selected, the mean of this sample belongs to a sampling distribution.

a. What is the shape of this distribution?

b. What is the mean of this sampling distribution?

c. What is the standard deviation of this sampling distribution?

2. More Americans heat their homes with natural gas than any other fuel. According to the American Gas Association, the national average price of natural gas sold to residential customers in 1997 was 62 cents per therm, about 18 percent less than it cost ten years earlier, in inflation-adjusted dollars. Source: 1997 Gas Facts, American Gas Association.

If the standard deviation for prices of natural gas sold to residential customers is 11 cents per therm and a random sample of 200 residential customers in 1997 is selected, the mean of this sample belongs to a sampling distribution.

a. What is the shape of this sampling distribution?

b. What is the mean of this sampling distribution?

c. What is the standard deviation of this sampling distribution?

3. According to the 1993 World Factbook, the 1993 total fertility rate (mean number of children born per woman) for Madagascar is 6.75. Suppose the standard deviation of the total fertility rate is 2.5. The mean number of children for a sample of 200 randomly selected women is one value of many that form the sampling distribution of sample means.

a. What is the mean value for this sampling distribution?

b. What is the standard deviation of this sampling distribution?

c. Describe the shape of this sampling distribution.

4. According to the 1994 World Almanac, the average speed of winds in Honolulu, Hawaii, equals 11.4 miles per hour. Assume that wind speeds are approximately normally distributed with a standard deviation of 3.5 miles per hour.

a. Find the probability that the wind speed on any one reading will exceed 13.5 miles per hour.

b. Find the probability that the mean of a random sample of 9 readings exceeds 13.5 miles per hour.

c. Do you think the assumption of normality is reasonable? Explain.

d. What effect do you think the assumption of normality had on the answers to (a) and (b)? Explain.

5. According to the U.S. Dept. of Energy, the average price of unleaded regular gasoline sold at service stations throughout the nation in 1996 was $1.23 per gallon. Assume that gasoline prices in general are normally distributed with a standard deviation of $.16 per gallon.

Source: Energy Administration, U.S. Dept. of Energy, Monthly Energy Review, June 1997.
A random sample of 45 stations in 1996 is selected and the pump prices for unleaded regular gasoline are recorded. Find the probability that the sample mean price:

a. exceeds $1.28 per gallon.

b. is less than $1.19 per gallon.

c. is between $1.20 and $1.27 per gallon.

6. According to the World Almanac and Book of Facts - 1994, the median weekly earnings of full-time wage and salary women, age 16 years or older in 1992, equals $381. Assume that the wages and salaries are normally distributed with  = $85.

a. Find the probability that the mean weekly earnings of a sample of 250 such women is between $375 and $385, if the mean equals $381.

b. Do you think the assumption of normality is reasonable? Explain.

c. What effect do you think the assumption of normality about the x distribution had on the answer to (a)? Explain.

d. Do you think the assumption of mean equals $381 is reasonable? Explain.

e. What effect do you think the assumption about the value of the mean had on the answer to (a)? Explain.

7. According to the August 1994 issue of Employment and Earnings, the June 1994 average weekly earnings for employees in general automotive repair shops was $406.15. Suppose the standard deviation for the weekly earnings for such employees is $55.50. Assuming that this mean and standard deviation are the current values, find the following probabilities for the mean of a sample of 100 such employees.

a. The probability the mean of the sample is less than $400.

b. The probability the sample mean is between $400 and $410.

c. The probability the mean of the sample is greater than $415.

d. Explain why the assumption of normality about the x distribution was not involved in the solution to (a), (b), and (c).

8. The Gallup Poll has been surveying the public for many years. When repeated sampling is used to track America's attitudes, the sample statistic reported, the percentage of yes responses, does not form a sampling distribution, but rather it forms a time series and demonstrates a trend. (Time series is a topic not covered in this text; however, many of its components are.) Complete the following questions to help recognize and understand the difference between repeated samples that belong to a sampling distribution and those that belong to a time series.

Year	Yes	No
1937	34	66
1949	48	48
1955	52	44
1967	57	39
1969	54	39
1971	66	29
1975	73	23
1978	76	19
1983	80	16
1984	78	17
1987	82	12
2000	xx	zz

a. Plot a scatter diagram displaying the Selected National Trend information (above), using the year as the input variable and the percentage of yes responses as the output variable y.

b. On the scatter diagram drawn in (a), plot the percentages of the no responses as a second output variable using the year number as the input variable x.

c. Do you see what could be called a trend? Explain.

d. Make a prediction for the percentage of Americans who would vote for a woman president. How did you use the cases study information?

e. Sampling distributions involve repeated sampling from the same population, but with a completely different purpose. Explain, in your own words, how a sampling distribution is different than the chapter case study illustration.

f. Repeated sampling, like that in the chapter case study and that used in quality control, is carried out for the purpose of "tracking" the statistic being studied. Describe, in your own words, the purpose of studying a statistic from repeated samples as a sampling distribution.

9. According to an article in Pharmaceutical News (January 1991), a person age 65 or older will spend, on the average, $300 on personal-care products per year. If we assume that the amount spent on personal-care products by individuals 65 or older is normally distributed and has a standard deviation equal to $75, what is the probability that the mean amount spent by 25 randomly selected such individuals will fall between $250 and $350?

10. A report in Newsweek (November 12, 1990) stated that the day-care cost per week in Boston is $109. If this figure is taken as the mean cost per week and if the standard deviation were known to be $20, find the probability that a sample of 50 day-care centers would show a mean cost of $100 or less per week.