As part of the recruitment of new businesses, the city’s economic development ­department wants to estimate the gross profit margin of small businesses ( under $ 1 million in sales) currently residing in the city. A random sample of the previous year’s annual reports of 15 small businesses shows the mean net profit margin to be 7 .2% ( of sales) with a standard deviation of 12.5%.
a. Construct a 99% confidence interval for the mean gross profit margin of μ of all small businesses in the city.
b. The city manager reads the report and states that the confidence interval for m constructed in part (a) is not valid because the data are obviously not normally distributed and thus the sample size is too small. Based on just knowing the mean and standard deviation of the sample of 15 businesses, do you think the city manager is valid in his conclusion about the data? Explain your answer.