The Washington Capitals recently won the National Hockey League's Stanley Cup championship, and there was a parade on June 12 to celebrate. Let's say...

The Washington Capitals recently won the National Hockey League's Stanley Cup championship, and there was a parade on June 12 to celebrate. Let's say 500,000 people attended the parade. We want to interview (sample) some of the Capitals fans at the parade to find things out about Capitals fans in general.

1)     Suppose we believe that the standard deviation of Capitals fans' income is $20,000 (close to the overall standard deviation of American incomes). We sample 45 Capitals fans at the parade (some people declined to give a response, but 45 people did give answers) and find that these 45 fans have a mean income of $58,500.

a.     Assuming this is a valid, random sample, specify a 95% confidence interval of what we think the mean of the population is given this sample. Please justify which statistic you use (t or z).

b.     One Capitals fan, during the interviews, says he thinks that the mean income of Capitals fans is $61,000. Does our sample (again, let's assume it's a valid sample), support his belief?

c.      If the fan from part b is correct, what is the probability of a sample of 45 Caps fans having a mean income less than or equal to $58,500?  

d.     How many fans do we need to interview to be 95% confident that we have a sample mean within $2,000 of the population mean.

e.     Explain two ways that this sampling method is biased (remember that we are trying to estimate the mean income of all Capitals fans) - you don't necessarily need to name the type of bias involved, but just explain two things wrong with it.

2)     We also ask those 45 people whether they have season statistics to estimate what percentage of Capitals fans actually have season tickets. (Again, assume our sampling method is valid for questions 2 and 3)

a.     Suppose 15 of the interviewees say they have season tickets. Based on this sample, specify a 99% confidence interval for the percentage of Caps fans who have season tickets.  

b.     How many fans would we need to interview to be 99% confident that we have an estimate of this proportion within plus or minus 3%?

3)     At the end of the parade, we decide that we want to ask some fans about their weights because we think that hockey fans are bigger than most people. Since we decided this so late, we only had time to interview 20 people. Among these 20, we found a mean weight of 215 pounds, with a standard deviation of 20 pounds, and we assume that the weights are normally distributed. Specify a 95% confidence interval for the mean weight of the population. Please justify which statistic you use (t or z).