Bob is interested in studying whether the average household income in city A is lower than the national benchmark, μ0 = $51,500. He collects income information from a random sample of 100 households, conducts a one-sample z test using the mean income of the sample the national benchmark, and constructs a confidence interval of the sample mean.
(a) Explain, in the context of the above scenario, what a sampling distribution is?
(b) It is well known that the distribution of income is highly and positively skewed, but the one-sample z test is based on critical values obtained from a standard normal distribution. Why is the skewness of income not a problem in this situation?
(c) Bob obtains a 95% confidence interval of [42,050, 55,125]. Interpret this confidence interval.
(d) What could Bob have done to obtain a narrower (i.e., more precise) confidence interval?