As part of the recruitment of new businesses in their city, the economic development department of the city wants to estimate the gross profit margin of small businesses (under one million dollars in sales) currently residing in their city. A random sample of the previous years annual reports of 15 small businesses shows the mean net profit margins 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 m 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.