Test Review (BVD 19 – 21)
One proportion inference
1) The countries of Europe report that 46% of the labor force is female. The United Nations wonders if the percentage of females in the labor force is the same in the United States. Representatives from the United States Department of Labor plan to check a random sample of over 10,000 employment records on file to estimate a percentage of females in the United States labor force.
A) The representative from the Department of Labor want to estimate a percentage of females in the United States labor force to within ± 5%, with 90% confidence. How many employment records should they sample?
B) They actually select a random sample of 525 employment records, and find that 229 of the people are females. Create the 90% confidence interval.
C) Interpret the confidence interval in this context.
D) Explain what 90% confidence means in this context.
E) Should the representatives from the Department of Labor conclude that the percentage of females in their labor force is lower than Europe’s rate of 46%? Explain.
2. A report on health care in the US said that 28% of Americans have experienced times when they haven’t been able to afford health care. A news organization randomly sampled 801 black Americans, of whom 38% reported that there had been times in the last year when they had not been able to afford medical care. Does this indicate that this problem is more severe among black Americans?
A) Test and appropriate hypothesis, showing all steps, and stating your conclusion.
B) Explain what your p-value means in this context.
3. A 1996 report from the U.S. Consumer Product Safety Commission claimed that at least 90% of all American homes have at least one smoke detector. The City of Grapevine Fire Department has been running a public safety campaign about smoke detectors consisting of posters, billboards, and ads on radio, and TV and in the newspaper. The city wonders if this concerted effort has raised the local level above the 90% national rate. Building inspectors visit 400 randomly selected homes and find that 376 have smoke detectors. Is this strong evidence that the local rate is higher than the national rate? Be sure to show all steps.
4. The owner of a small clothing store is concerned that only 28% of people who enter her store actually buy something. A marketing salesman suggests that she invest in a new line of celebrity mannequins (picture Adam Sandler modeling the latest jeans…) He loans her several different “celebrities” to scatter around the store for a two-week trial period. The owner carefully counts how many shopper enter the store and how many buy something so that at the end of the trial period she can decide if she’ll purchase the mannequins.
A) Write the owners null and alternative hypotheses.
B) In this context describe a Type I error and the impact such an error would have on the store.
C) In this context describe a Type II error and the impact such an error would have on the store.
D) Based on data that she collected during the trial period the store’s owner found that a 98% confidence interval for the proportion of all customers who might buy something was (27%, 35%). What conclusion could she reach about the mannequins based on this interval?
E) Describe to the owner an advantage and a disadvantage of using a confidence level of 90% instead.
F) Over the trial period, the rate of in-store sales rose to 30% of shoppers. Even though the owner is convinced that the proportion of buyers rose from 28% to 30%, why might she still choose not to buy the mannequins?
5. Your local newspaper polls a random sample of 330 voters, finding 144 who say they will vote “yes” on the upcoming school budget. Create and interpret a 95% confidence interval for the true proportion of voters who will vote “yes” on the upcoming school budget.
6. Traffic accidents are the leading cause of death among people aged 15 to 20. In May 2002, the National Highway Traffic Safety Administration reported that even though only 6.7% of licensed drivers are between 15 & 20 years old, they were involved in 14.3% of all fatal crashes. Insurance companies have long known that teenage boys were high risks, but what about teenage girls? One insurance company found that the driver was a teenage girl in 44 of the 388 fatal accidents they investigated. Is this strong evidence that the fatal accident rate is lower for teenage girls than the 14.3% for teens in general?
7. In may of 2002, the Gallup Organization asked a random sample of 527 American adults this question:
“If you could choose between the following two approaches, which do you think is the better penalty for murder, the death penalty or life imprisonment, with absolutely no possibility of parole?”
Of those polled, 52% chose the death penalty.
A) Create and interpret a 95% confidence interval for the percentage of all American adults who favor the death penalty.
B) Based on your confidence interval, is it clear that the death penalty has majority support? Explain.
C) If pollsters wanted the follow up on this poll with another survey that could determine the level of support for the death penalty to within 2% for 95% confidence, how many people should they poll?
8. An advertising agency won’t sign an athlete to do product endorsements unless it is sure the person is known to more than 25% of its target audience. The agency always conducts a poll of 500 people to investigate the athlete’s name recognition before offering a contract. Then it tests H0: p = 0.25 against Ha: p > .25 at a 5% significance level.
A) Why does the company use upper tail tests in this situation?
B) Explain what Type I and Type II errors would represent in this context, and describe the risk that each error poses to the company.
C) The company is thinking of changing its test to use a 10% level of significance. How would this change the company’s exposure to each type of risk?