Question

Mrs. N is the manager of a mall. She is reading a report, prepared by the mall’s accounting staff which outlines various strategies for increasing mall’s income. Included in the report are results from a study on how much customers spend at small kiosks while in the mall. She reads about a “95% confidence interval for the mean spending per customer as being “$20 ± $5”. a. Give an interpretation of this interval in the context of this problem. b. Find two wrong interpretations of the 95% confidence interval Mrs. N. read in her report. And explain in what way those interpretations are wrong. (Make sure that the mistakes in the interpretations are different in nature. Similar natured ones will be counted as one.) c. While reading the report, she wonders why she has only 95% confidence interval rather than 100% confidence interval. Explain to Mrs. N. why a 100% confidence interval is not useful.

Answer 1

a. Interpretation : We are 95% confident in stating that the actual population mean spending per customer at small kiosks lies in the range 15$ to 25$.

b. Mistake 1 : There is 95% chance that the true population mean spending is inside the interval (15,25). The reason why this is wrong is that the Population mean is a fixed constant, there shouldn't be any probabilistic notion associated with its existence being in some interval. The chance or randomness aspect comes from our lack of full information availability.

Mistake 2 : 95% of all (population ) customer spending amount at small kiosk fall within the interval (15,25) . This is wrong because confidence interval is computed from a specific finite sample taken from the population. So a 95% confidence range obtained from a particular sample from large population doesnt necessarily include inside it 95% of all population data.

c. If we want 100% confidence interval, then it would be ( - infinity, + infinity ) . Which is not of much use because we already know that mean spending will be a finite number. Higher the confidence level one wants, one has to compromise by relaxing width to a larger value.