Table 6.4 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.

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

b. Construct a histogram.

c. Draw a smooth curve through the midpoints of the tops of the bars of the histogram.

\(X\) is a random variable denoting the maximum capacity of sports stadiums.

Number of samples, \(n=60\)

The required sample mean is:

\(\bar{x}=\frac{1}{n}\sum_{i-1}^{n}x_{i}=\frac{3608185}{60}=60136\)

The required sample standard deviation is:

\(s=\sqrt{\frac{1}{n-1}\sum_{i-1}^{n}(x_{i}-\bar{x})^{2}}=\sqrt{\frac{1}{60-1}*6465289336}=10468\)

Therefore we conclude that the sample mean is \(60,136\) and the sample standard deviation is \(10,468\).

To construct a histogram, first the data is converted into a continuous frequency distribution. The frequency distribution is obtained as given below:

Using this table, the required histogram can easily be constructed by plotting the maximum capacity on the \(x-\)axis and the number of stadiums on the \(y-\)axis. The histogram obtained is:

From the histogram, it can be easily seen that maximum number of stadiums have a capacity of \(50,000–60,000\) spectators and very few stadiums have the capacity of \(80,000–90,000\) spectators.

To construct a curve through the midpoints of the histogram, first a histogram is constructed using the continuous frequency distribution which is given as:

Using this table, the required curve is obtained by plotting the midpoints on the \(x-\)axis and the number of stadiums on the \(y-\)axis. The curve obtained is:

The curve shows an increase from \(40,000–50,000\) to \(50,000–60,000\) and decreases thereafter. Also, it can be easily seen that maximum number of stadiums have the capacity of \(50,000–60,000\) spectators.