Using the Normal Distribution · Exercise 33

shows a sample of the maximum capacity—maximum number of spectators—of sports stadiums. The table does not include horse-racing or motor-racing stadiums.
$$\begin{array}{cccccc}\text{40,000}& \text{40,000}& \text{45,050}& \text{45,500}& \text{46,249}& \text{48,134}\\ \text{49,133}& \text{50,071}& \text{50,096}& \text{50,466}& \text{50,832}& \text{51,100}\\ \text{51,500}& \text{51,900}& \text{52,000}& \text{52,132}& \text{52,200}& \text{52,530}\\ \text{52,692}& \text{53,864}& \text{54,000}& \text{55,000}& \text{55,000}& \text{55,000}\\ \text{55,000}& \text{55,000}& \text{55,000}& \text{55,082}& \text{57,000}& \text{58,008}\\ \text{59,680}& \text{60,000}& \text{60,000}& \text{60,492}& \text{60,580}& \text{62,380}\\ \text{62,872}& \text{64,035}& \text{65,000}& \text{65,050}& \text{65,647}& \text{66,000}\\ \text{66,161}& \text{67,428}& \text{68,349}& \text{68,976}& \text{69,372}& \text{70,107}\\ \text{70,585}& \text{71,594}& \text{72,000}& \text{72,922}& \text{73,379}& \text{74,500}\\ \text{75,025}& \text{76,212}& \text{78,000}& \text{80,000}& \text{80,000}& \text{82,300}\end{array}$$

(a) Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums.
(b) Construct a histogram.
(c) Draw a smooth curve through the midpoints of the tops of the bars of the histogram.
(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 the maximum capacity of sports stadiums is less than 67,000 spectators.
(g) Determine the cumulative relative frequency that the maximum capacity of sports stadiums is less than 67,000 spectators. Hint—Order the data and count the sports stadiums that have a maximum capacity less than 67,000. Divide by the total number of sports stadiums in the sample.
(h) Why aren’t the answers to part f and part g exactly the same?

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

OpenStax, Statistics · §6.2 Using the Normal Distribution · Exercise 33