Statistics Exercise Questions

"Probability":

1) The U.S. National Highway Traffic Safety Administration gathers data concerning the causes of highway crashes where at least one fatality has occurred. The following probabilities were determined from the 1998 annual study (BAC is blood-alcohol content).

Source: Statistical Abstract of the United States, 2000, Table 1042.

P(BAC = 0 0 Crash with fatality) = .616

P(BAC is between .01 and .09 0 Crash with fatality) = .300

P(BAC is greater than .09 0 Crash with fatality) = .084

Over a certain stretch of highway during a 1-year period, suppose the probability of being involved in a crash that results in at least one fatality is .01. It has been estimated that 12% of the drivers on this highway drive while their BAC is greater than .09. Determine the probability of a crash with at least one fatality if a driver drives while legally intoxicated (BAC greater than .09).

2) Your favorite team is in the final playoffs. You have assigned a probability of 60% that it will win the championship. Past records indicate that when teams win the championship, they win the first game of the series 70% of the time. When they lose the series, they win the first game 25% of the time. The first game is over; your team has lost. What is the probability that it will win the series?

"Introduction to Estimation":

1) a. A random sample of 25 was drawn from a normal distribution with a standard deviation of 5. The sample mean is 80. Determine the 95% confidence interval estimate of the population mean.

b. Repeat part (a) with a sample size of 100.

c. Repeat part (a) with a sample size of 400.

d. Describe what happens to the confidence interval estimate when the sample size increases.

2) a. A statistics practitioner randomly sampled 100 observations from a population with a standard deviation of 5 and found that x̄ is 10. Estimate the population mean with 90% confidence.

b. Repeat part (a) with a sample size of 25.

c. Repeat part (a) with a sample size of 10.

d. Describe what happens to the confidence interval estimate when the sample size decreases.

"Introduction to hypothesis testing":

1) A statistics practitioner wants to test the following hypotheses with σ = 20 and n = 100:

H0: μ = 100

H1: μ > 100

a. Using a = .10 find the probability of a Type II error when μ = 102.

b. Repeat part (a) with a = .02.

c. Describe the effect on β of decreasing a

2) a. Calculate the probability of a Type II error for the following hypotheses when μ = 37:

H0: μ = 40

H1: μ < 40

The significance level is 5%, the population standard deviation is 5, and the sample size is 25.

b. Repeat part (a) with a = 15%.

c. Describe the effect on β of increasing a.

"Inference about a population":

1) With gasoline prices increasing, drivers are more concerned with their cars’ gasoline consumption. For the past 5 years a driver has tracked the gas mileage of his car and found that the variance from fill-up to fill-up was σ2 = 23 mpg2. Now that his car is 5 years old, he would like to know whether the variability of gas mileage has changed. He recorded the gas mileage from his last eight fill-ups; these are listed here. Conduct a test at a 10% significance level to infer whether the variability has changed.

2) During annual checkups physicians routinely send their patients to medical laboratories to have various tests performed. One such test determines the cholesterol level in patients’ blood. However, not all tests are conducted in the same way. To acquire more information, a man was sent to 10 laboratories and had his cholesterol level measured in each. The results are listed here. Estimate with 95% confidence the variance of these measurements.