Using the Normal Distribution · Exercise 31

Thuy Dau, Ngoc Bui, Sam Su, and Lan Voung conducted a survey as to how long customers at Lucky claimed to wait in the checkout line until their turn. Let X = time in line. displays the ordered real data (in minutes):
$$\begin{array}{ccccc}\text{0.50}& \text{4.25}& \text{5}& \text{6}& \text{7.25}\\ \text{1.75}& \text{4.25}& \text{5.25}& \text{6}& \text{7.25}\\ \text{2}& \text{4.25}& \text{5.25}& \text{6.25}& \text{7.25}\\ \text{2.25}& \text{4.25}& \text{5.5}& \text{6.25}& \text{7.75}\\ \text{2.25}& \text{4.5}& \text{5.5}& \text{6.5}& \text{8}\\ \text{2.5}& \text{4.75}& \text{5.5}& \text{6.5}& \text{8.25}\\ \text{2.75}& \text{4.75}& \text{5.75}& \text{6.5}& \text{9.5}\\ \text{3.25}& \text{4.75}& \text{5.75}& \text{6.75}& \text{9.5}\\ \text{3.75}& \text{5}& \text{6}& \text{6.75}& \text{9.75}\\ \text{3.75}& \text{5}& \text{6}& \text{6.75}& \text{10.75}\end{array}$$

(a) Calculate the sample mean and the sample standard deviation.
(b) Construct a histogram.
(c) Draw a smooth curve through the midpoints of the tops of the bars.
(d) In words, describe the shape of your histogram and smooth curve.
(e) Let the sample mean approximate μ and the sample standard deviation approximate σ. The distribution of X can then be approximated by X ~ _____(_____,_____)
(f) Use the distribution in part e to calculate the probability that a person will wait fewer than 6.1 minutes.
(g) Determine the cumulative relative frequency for waiting less than 6.1 minutes.
(h) Why aren’t the answers to part f and part g exactly the same?
(i) Why are the answers to part f and part g as close as they are?
(j) If only ten customers has been surveyed rather than 50, do you think the answers to part f and part g would have been closer together or farther apart? Explain your conclusion.

This is an open-ended exercise: its source gives no single answer.

OpenStax, Introductory Statistics 2e · §6.2 Using the Normal Distribution · Exercise 31