Question

A sample of the maximum capacity of spectators of sports stadiums is included in the table. 40,000 49,133 51,500 52,692 55,000 59,680 62,872 66,161 70,585 75,025 40,000 50,071 51,900 53,864 55,000 60,000 64,035 67,428 71,594 76,212 45,050 50,096 52,000 54,000 55,000 60,000 65,000 68,349 72,000 78,000 45,500 50,466 52,132 55,000 55,082 60,492 65,050 68,976 72,922 80,000 46,249 50,832 52,200 55,000 57,000 60,580 65,647 69,372 73,379 80,000 48,134 51,500 52,530 55,000 58,008 62,380 66,000 70, 107 74,500 82,300 a. Calculate the sample mean and the sample standard deviation for the maximum capacity of sports stadiums (see data above). 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? And the sample standard deviation approximate? 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. h. Why aren't the answers to part f and part g exactly the same?

Answer 1

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

a. Mean = 60143

Standard Deviation = 10462

b.

Histogram of Max. Cap 12 10 8 Frequency 2 0 50000 70000 80000 CS Sca40000 with CamScanner 60000 Max. Сар
c. The smooth curve over histogram
Histogram of Max. Cap Largest Extreme Value 0.00004- LOC 55111 Scale 9160 N 60 0.00003 Density 0.00002 0.00001 80000 90000 0.00000 CS Scan 40000ith 50000 CamScanner 60000 70000 Max. Cap

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.