1. A commercial bank is concerned about its image among its clients. In a random survey that was run last year among 500 clients, 54% said that they were satisfied with the level of customer service. This year, in another survey among 400 clients, only 45% said that they were satisfied with the level of customer service.
a. At the 5% level of significance, can we infer that the level of satisfaction among the bank's clients has dropped?
b. At the 5% level of significance, can we infer that the level of satisfaction among the bank's clients has decreased by more than 5 percentage points?
2. There is a lot of controversy with regard to the ills of gambling. The state lottery office claims that the average household income of those people playing at the casino is $40,000 or more. Assume that the distribution of household income of those people playing at the casino is normally distributed with a standard deviation of $6,800. Suppose that for a sample of 25 households, it is found that the average income was $38,200.
a. Is the state lottery incorrect in asserting that the average household income of casino players exceeds $40,000 at 10% level of significance?
b. What should the level of significance be to arrive at a conclusion that would be the opposite of the one you have arrived at (a) above?
c. Based on the sample findings, estimate the true mean income of casino players with 90% probability.
d. If we wanted to shorten the interval of the estimation of the true mean income of casino players to a width of $1000 in total, what should the size of our sample be to achieve it?
e. If the true average income of those playing in the casino were actually $37,500, what would be the probability of accepting the (false) null hypothesis in (a) above?
3. The following data represent weights (in pounds) for two random samples of men of approximately 5 feet 10 inches tall and of medium build. The only difference is that the first group is comprised of athletic persons and the second of non-athletic ones. Weights are assumed to follow normal distributions.
Athletic men: 152, 148, 156, 155, 157, 162, 159, 168, 150, 173.
Non-athletic men: 155, 157, 169, 170, 171, 161, 181, 165, 183.
a. Calculate the means and variances of the two samples.
b. Can we conclude at the 5% significance level that the population mean weight of athletic men is lower than the population mean weight of non-athletic men?
c. At the 5% level of significance, can we infer that the population mean weight of athletic men is lower than the population mean weight of non-athletic men by more than 2 pounds?
d. Estimate with 90% confidence the difference in the mean weight between the two groups.
e. After having compared the means of the weights of the two groups, can we conclude at the 5% significance level that the variance of the weights of the non-athletic men is larger that the variance of the weights of the athletic men?