A firm conducted an extensive survey of a town with a population of only 10,000.

2. A firm conducted an extensive survey of a town with a population of only 10,000. The sample size of this survey was 1,000 (which is 10% of the population of the town), and the firm’s statistics expert says that the sample size is “quite adequate” to get estimates needed. The firm then decides to extend the survey nationwide, but the CEO says it will be too expensive, since to maintain an “adequate” sample size, they will have to survey 320,000 people (this is about 10% of the population of the U.S.).

a) The statistician informs the CEO that there is no need to increase the sample size beyond 1,000 for the nationwide survey in order to get “adequate” estimates. How can this be?

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b) The CEO then says that if 1,000 surveys is adequate for a nationwide sample, they could just use the survey from the original sample from the small town. The statistician says this would be inappropriate, and a new sample must be conducted. Why?

1. Compute the following probabilities:

a) P(t10 greater than or equal to 1.75) where t10 has a t distribution with 10 degrees of freedom.

b) P(t100 greater than or equal to 1.75) where t100 has a t distribution with 100 degrees of freedom. Compare to part a). How do you explain the difference between this result and the one obtained in part a).

c) P(Z greater than or equal to 1.75) where Z is a standard normal random variable. Compare to parts a) and b). How do you explain the differences in these probabilities?

2. If, based on a sample of 500, a political candidate finds that 245 people would vote for him in a two-person race, what is the 90% confidence interval for his expected proportion of the vote? How would you explain your results to the candidate?

3. Using the data in the worksheet Consumer Transportation Survey.xls, develop and interpret 95% confidence intervals for:

a) the mean hours per week that individuals spend in their vehicles

b) the average number of miles driven per week

c) the proportion of individuals who are satisfied with their vehicle

d) the proportion of individuals who have at least one child

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