Probability And Statistics Discussion Question
“Probability”: Exercises 6.75 and 6.81 “Introduction to Estimation”: Exercises 10.11 and 10.14 “Introduction to hypothesis testing”: Exercises 11.52 and 11.60 “Inference about a population”: Exercises 12.73 and 12.74
6.75 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).
6.81 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?
10.11 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.
10.14 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.
11.52 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
· 11.60 Suppose that in Example 11.1 we wanted to determine whether there was sufficient evidence to conclude that the new system would not be costeffective. Set up the null and alternative hypotheses and discuss the consequences of Type I and Type II errors. Conduct the test. Is your conclusion the same as the one reached in Example 11.1 ? Explain. (the following goes with this question)
EXAMPLE 11.1 Department Store’s New Billing System
The manager of a department store is thinking about establishing a new billing system for the store’s credit customers. After a thorough financial analysis, she determines that the new system will be cost-effective only if the mean monthly account is more than $170. A random sample of 400 monthly accounts is drawn, for which the sample mean is $178. The manager knows that the accounts are approximately normally distributed with a standard deviation of $65. Can the manager conclude from this that the new system will be cost-effective?
This example deals with the population of the credit accounts at the store. To conclude that the system will be cost-effective requires the manager to show that the mean account for all customers is greater than $170. Consequently, we set up the alternative hypothesis to express this circumstance:
H1: μ > 170 (Install new system)
If the mean is less than or equal to 170, then the system will not be cost-effective. The null hypothesis can be expressed as
H0: μ ≤ 170 (Do not install new system)
However, as was discussed in Section 11-1, we will actually test μ = 170, which is how we specify the null hypothesis:
H0: μ = 170
As we previously pointed out, the test statistic is the best estimator of the parameter. In Chapter 10, we used the sample mean to estimate the population mean. To conduct this test, we ask and answer the following question: Is a sample mean of 178 sufficiently greater than 170 to allow us to confidently infer that the population mean is greater than 170?
There are two approaches to answering this question. The first is called the rejection region method. It can be used in conjunction with the computer, but it is mandatory for those computing statistics manually. The second is the p-value approach, which in general can be employed only in conjunction with a computer and statistical software. We recommend, however, that users of statistical software be familiar with both approaches.
12.73 Xr12-73 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.
12.74 Xr12-74 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.