BackConstruction 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.
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)?
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)?
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.
Net worth is defined as total assets (value of house, cars, money, etc.) minus total liabilities (mortgage balance, credit card debt, etc.). According to a recent study by TNS Financial Services, 7% of American households had a net worth in excess of \$1 million (excluding their primary residence). A random sample of 1000 American households results in 82 having a net worth in excess of \$1 million. Explain why the results of this survey do not necessarily imply that the proportion of households with a net worth in excess of \$1 million has increased.
What are the mean and standard deviation of the sampling distribution of x̄? What are the mean and standard deviation of the sampling distribution of p̂?
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.
b. Would it be unusual for a player who is a career 0.280 hitter to have a season in which he hits at least 0.310?
Describe the circumstances under which the shape of the sampling distribution of p̂ is approximately normal.
According to the National Center for Health Statistics, 22.4% of adults are smokers. A random sample of 300 adults is obtained.
a. Describe the sampling distribution of p̂, the sample proportion of adults who smoke.
True or False: The population proportion and sample proportion always have the same value.
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 = 1460 is obtained from a population whose size is N = 1,500,000 and whose population proportion with a specified characteristic is p = 0.42.
c. What is the probability of obtaining x = 584 or fewer individuals with the characteristic?
Reincarnation Suppose 21% of all American teens (age 13–17 years) believe in reincarnation.
b. Explain why Bob’s sample of 100 randomly selected American teens might result in 18 who believe in reincarnation, while Alicia’s independent sample of 100 randomly selected American teens might result in 22 who believe in reincarnation.
Reincarnation 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?