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Answer :
a. To compute the confidence interval, use a t-distribution.
b. With 90% confidence, the population mean number of pounds per person per week is between 32.382 pounds and 35.418 pounds.
c. If many groups of 196 randomly selected members are studied, approximately 90% of these confidence intervals will contain the true population mean number of pounds of trash generated per person per week, while about 10% will not contain the true population mean number of pounds of trash generated per person per week.
(a) To compute the confidence interval, we use a t-distribution since the sample size is less than 30 and the population standard deviation is unknown.
(b) With 90% confidence, the population mean number of pounds per person per week is between [32.674, 35.126] pounds. Here, we use the formula for the confidence interval:
CI = xbar ± (t * (s / √n)),
where xbar is the sample mean, s is the sample standard deviation, n is the sample size, and t is the critical value from the t-distribution corresponding to the desired confidence level.
Using the given information, the lower bound of the confidence interval is 33.9 - (1.645 * (8.3 / √196)) ≈ 32.674, and the upper bound is 33.9 + (1.645 * (8.3 / √196)) ≈ 35.126.
(c) If many groups of 196 randomly selected members are studied, approximately 90% of these groups' confidence intervals will contain the true population mean number of pounds of trash generated per person per week, while approximately 10% will not contain the true population mean. This means that in repeated sampling, about 90% of the calculated confidence intervals will capture the actual population mean, providing a measure of accuracy for the estimation process. The remaining 10% will not include the true population mean, representing the possibility of estimation error or uncertainty in those particular intervals.
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