Note: since this is part of exam 3, you are to work individually on these problems. Do not discuss these questions with anyone except Dr. Williams. Work each problem on a separate page, bring the take home problems (and all intermediate calculations) with you to the final exam on Thursday, May 14. One or more of these problems will be handed in with the final exam, and information from others will be used on the in-class questions.
1. A statistician has been challenged to estimate the number of gumballs in a large container. The gumballs in the container are all red, blue, or yellow. The statistician took a simple random sample of 300 gumballs, and replaced them with purple gumballs that were the same size and weight as the original gumballs. After stirring the container well, a second simple random sample of 500 gumballs is taken, of which 37 are purple.
a) Estimate the number of gumballs in the container, and compute a bound for your estimate.
b) If you were planning a future study to estimate the number of gumballs in a similar container, about how many gumballs should you take out originally, and how many should you later sample in order to achieve a bound that is less than or equal to 1000?
2. An investigator wants to estimate the proportion of Washington state residents that are under age 18. Of the 39 counties in the state of Washington, she takes a random sample of 10 counties, and is able to obtain the following Census information for those counties:
County Population Percent under age 18 Benton 159,414 26.5 Columbia 3,986 19.2 Franklin 69,578 33.6 Island 81,326 22.1 Kittitas 38,542 18.2 Mason 56,384 21.1 Pierce 773,165 25.4 Snohomish 676,898 25.2 Wahkiakum 4,039 17.8 Yakima 233,062 30.6
a) Based on these data, estimate the proportion of Washington state residents who are under age 18. Compute a bound for your estimate.
b) Based on these results, for a future study to estimate the proportion of Washington state residents that are under age 18, how many counties should be sampled to have a bound of B=.03?
3. To estimate the average salary of players in the National Basketball Association (NBA) for the 2008-2009 season, a sample was taken. First, a simple random sample of 7 teams (out of 32 teams) was taken. For those 7 teams, a 1-in-3 systematic random sample of players was taken and the salary of selected players was recorded. The data from the sample are available in this file.
Using these data, calculate an estimate of the average salary of NBA players in the 2008-2009 season, and compute a bound for your estimate.
4. On the question above about counties in Washington state, if we performed a study to estimate the number of unemployed adults in Washington, are there potential disadvantages to the sampling method used in question 2? Suggest one or more alternative sampling methods to use county-level data for estimation of the number of unemployed adults in Washington state, and briefly explain why your method would be better.