Voter Preference A political candidate estimates that 30%

Explain why a normal distribution can be used as an approximation to a binomial distribution. What conditions must be met to use the normal distribution to approximate the binomial distribution? Why is a correction for continuity necessary?

(ans) Use the normal approximation to the binomial to find the probabilities for the specific value(s) of X. a. n _ 30, p _ 0.5, X _ 18 0.0811 b. n _ 50, p _ 0.8, X _ 44 0.0516 c. n _ 100, p _ 0.1, X _ 12 0.1052 d. n _ 10, p _ 0.5, X _ 7 0.1711 e. n _ 20, p _ 0.7, X _ 12 0.2327 f. n _ 50, p _ 0.6, X _ 40 0.9988

Check each binomial distribution to see whether it can be approximated by a normal distribution (i.e., are np _ 5 and nq _ 5?). a. n _ 20, p _ 0.5 Yes d. n _ 50, p _ 0.2 Yes b. n _ 10, p _ 0.6 No e. n _ 30, p _ 0.8 Yes c. n _ 40, p _ 0.9 No f. n _ 20, p _ 0.85 No

School Enrollment Of all 3- to 5-year-old children, 56% are enrolled in school. If a sample of 500 such children is randomly selected, find the probability that at least 250 will be enrolled in school. 0.9970 Source: Statistical Abstract of the United States.

Youth Smoking Two out of five adult smokers acquired the habit by age 14. If 400 smokers are randomly selected, find the probability that 170 or fewer acquired the habit by age 14. 0.8577 Source: Harpers Index.

Mail Order A mail order company has an 8% success rate. If it mails advertisements to 600 people, find the probability of getting less than 40 sales.

Voter Preference A political candidate estimates that 30% of the voters in her party favor her proposed tax reform bill. If there are 400 people at a rally, find the probability that at least 100 voters will favor her tax bill. Based on your answer, is it likely that 100 or more people will favor the bill? 0.9875

Household Computers According to recent surveys, 60% of households have personal computers. If a random sample of 180 households is selected, what is the probability that more than 60 but fewer than 100 have a personal computer? Source: New York Times Almanac. 0.0984

Female Americans Who Have Completed 4 Years of College The percentage of female Americans 25 years old and older who have completed 4 years of college or more is 26.1. In a random sample of 200 American women who are at least 25, what is the probability that at most 50 have completed 4 years of college or more? Source: New York Times Almanac. 0.3936

Population of College Cities College students often make up a substantial portion of the population of college cities and towns. State College, Pennsylvania, ranks first with 71.1% of its population made up of college students. What is the probability that in a random sample of 150 people from State College, more than 50 are not college students? 0.0985 Source: www.infoplease.com

Elementary School Teachers Women comprise 80.3% of all elementary school teachers. In a random sample of 300 elementary teachers, what is the probability that less than three-fourths are women? 0.0087 Source: New York Times Almanac.

Telephone Answering Devices Seventy-eight percent of U.S. homes have a telephone answering device. In a random sample of 250 homes, what is the probability that fewer than 50 do not have a telephone answering device? 0.2005 Source: New York Times Almanac.

Parking Lot Construction The mayor of a small town estimates that 35% of the residents in the town favor the construction of a municipal parking lot. If there are 350 people at a town meeting, find the probability that at least 100 favor construction of the parking lot. Based on your answer, is it likely that 100 or more people would favor the parking lot? 0.9951; yes (TI: 0.9950)

Residences of U.S. Citizens According to the U.S. Census, 67.5% of the U.S. population were born in their state of residence. In a random sample of 200 Americans, what is the probability that fewer than 125 were born in their state of residence?

Recall that for use of a normal distribution as an approximation to the binomial distribution, the conditions np _ 5 and nq _ 5 must be met. For each given probability, compute the minimum sample size needed for use of the normal approximation.