BackReincarnation Suppose 21% of all American teens (age 13–17 years) believe in reincarnation.
f. What minimum sample size would you require in order for the distribution of the sample proportion to be modeled by the normal distribution?
Afraid to Fly According to a study conducted by the Gallup organization, the proportion of Americans who are afraid to fly is 0.10. A random sample of 1100 Americans results in 121 indicating that they are afraid to fly. Explain why this is not necessarily evidence that the proportion of Americans who are afraid to fly has increased since the time of the Gallup study.
Construction About 63% of the residents in a town are in favor of building a new high school. One hundred five residents are randomly selected. What is the probability that the sample proportion in favor of building a new school is less than 55%? Interpret your result.
Which of the following statements about sampling distributions for the sample mean is false?
MCAT The Medical College Admissions Test (MCAT) is used to help screen applicants to medical schools. Like many such tests, the MCAT uses multiple-choice questions with each question having five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of “a.”
b. Find the mean of the sampling distribution of the sample proportion.
Peanut and tree nut allergies are considered to be the most serious food allergies. According to the National Institute of Allergy and Infectious Diseases, roughly 1% of Americans are allergic to peanuts or tree nuts. A random sample of 1500 Americans is obtained.
a. Explain why a large sample is needed for the distribution of the sample proportion to be approximately normal.
"Getting Physical The figure shows the results of a survey of U.S. adults ages 18 to 29 who were asked whether they participated in a sport. In the survey, 48% of the men and 23% of the women said they participate in sports. The most common sports are shown below. Use this information in Exercises 29 and 30.
You randomly select 250 U.S. men ages 18 to 29 and ask them whether they participate in at least one sport. You find that 80% say no. How likely is this result? Do you think this sample is a good one? Explain your reasoning."
Variability in Baseball Suppose, during the course of a typical season, a batter has 500 at-bats. This means the player has the opportunity to get a hit 500 times during the course of a season. Further, suppose a batter is a career 0.280 hitter (he averages 280 hits every 1000 at-bats or he has 280 successes in 1000 trials of the experiment), so the population proportion of hits is 0.280.
d. Explain why a career 0.280 hitter could easily have a batting average between 0.260 and 0.300.
A simple random sample of size n = 75 is obtained from a population whose size is N = 10,000 and whose population proportion with a specified characteristic is p = 0.8.
b. What is the probability of obtaining x = 63 or more individuals with the characteristic? That is, what is P(p̂ ≥ 0.84)?
Hybridization A hybridization experiment begins with four peas having yellow pods and one pea having a green pod. Two of the peas are randomly selected with replacement from this population.
c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of peas with yellow pods? Does the mean of the sampling distribution of proportions always equal the population proportion?
Acceptance Sampling A shipment of 50,000 transistors arrives at a manufacturing plant. The quality control engineer at the plant obtains a random sample of 500 resistors and will reject the entire shipment if 10 or more of the resistors are defective. Suppose that 4% of the resistors in the whole shipment are defective. What is the probability the engineer accepts the shipment? Do you believe the acceptance policy of the engineer is sound?
What happens to the standard deviation of p̂ as the sample size increases? If the sample size is increased by a factor of 4, what happens to the standard deviation of p̂?
A simple random sample of size n = 1000 is obtained from a population whose size is N = 1,000,000 and whose population proportion with a specified characteristic is p = 0.35.
c. What is the probability of obtaining x = 320 or fewer individuals with the characteristic?
A simple random sample of size n = 200 is obtained from a population whose size is N = 25,000 and whose population proportion with a specified characteristic is p = 0.65.
b. What is the probability of obtaining x = 136 or more individuals with the characteristic? That is, what is P(p̂ ≥ 0.68)?
Describe the circumstances under which the shape of the sampling distribution of p̂ is approximately normal.