Question

Table 6.0 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.

40,000	40,000	45,050	45,500	46,249	48,134
49,133	50,071	50,096	50,466	50,832	51,500
51,500	51,900	52,000	52,132	52,200	52,530
52,692	53,864	54,000	55,000	55,000	55,000
55,000	55,000	55,000	55,082	57,000	58,008
59,680	60,000	60,000	60,492	60,580	62,380
62,872	64,035	65,000	65,050	65,647	66,000
66,161	67,428	68,349	68,976	69,372	70,107
70,585	71,594	72,000	72,922	73,379	74,500
75,025	76,212	78,000	80,000	80,000	82,300

Table 6.0

1. Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums (the data).

- sd = 60136 and 10468

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?

Answer 1

We find the Mean and standard deviation of the maximum seating capacity of stadiums using Minitab

a. Mean = 60143

Standard Deviation = 10462

b.

c. The smooth curve over histogram

d. The histogram is bell shaped, however slightly skewed to the right.

e. X can be more or less approximated by a normal distribution.

X ~ N(60143,10462)

f. P[ X < 67000]

= P[(X - 60143)/10462 < (67000 - 60143)/10462]

= P[ Z < 0.6554] {Z = (X - 60143)/10462 ~ N(0,1)}

= 0.74389 {Values obtained from a standard normal table}

g.

To determine the cumulative relative frequency that maximum capacity of sports stadiums is less than 67000 spectators.

= (Number of stadiums in sample with stadium capacity less than 67000)/ (Total number of stadiums in sample)

= 43/60

= 0.7167

h. The answers in f and g aren't exactly the same because in f we assume normality and fit a normal distribution to the data obtained and then calculate the probability. However in g we just calculate the exact probability based on sample values that are given in the data.