Part 5 of 5
Points: 0 of 1
The following data represent the number of games played in each series of an annual tournament from 1923 to 2019 . Complete parts (a) through (d) below.
\begin{tabular}{|c|c|c|c|c|}
\hline$x$ (games played) & 4 & 5 & 6 & 7 \\
\hline Frequency & 20 & 21 & 21 & 34 \\
\hline
\end{tabular}
(c) Compute and interpret the mean of the random variable $X$
\[
\mu_{\mathrm{x}}=5.7 \text { game(s) }
\]
(Round to one decimal place as needed.)
Interpret the mean of the random variable $\mathrm{X}$ Select the correct choice below and fill in the answer box within your choice.
(Round to one decimal place as needed)
A. The series, if played one time, would be expected to last about $\quad$ game(s).
B. The series, if played many times, would be expected to last about 5.7 game(s), on average.
(d) Compute the standard deviation of the random variable $\mathrm{X}$
\[
\sigma_{\mathrm{X}}=\square \text { game(s) }
\]
(Round to one decimal place as needed)
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Answer
Round the standard deviation to one decimal place. So, the standard deviation of the random variable X is approximately \(\boxed{1.2}\) games.
Step 1 ：Given data is the number of games played which is [4, 5, 6, 7] and their corresponding frequencies [20, 21, 21, 34]. The mean of the games played is given as 5.7.
Step 2 ：First, calculate the total number of values which is the sum of the frequencies. \(N = 20 + 21 + 21 + 34 = 96\).
Step 3 ：Next, calculate the variance. The variance is the sum of the squared differences between each value and the mean, multiplied by the frequency of that value, all divided by the total number of values. Variance = \(\frac{\sum ((x - \mu)^2 * f)}{N}\) where \(x\) is the value, \(\mu\) is the mean, \(f\) is the frequency of the value and \(N\) is the total number of values. Substituting the given values, we get Variance = \(\frac{(4-5.7)^2*20 + (5-5.7)^2*21 + (6-5.7)^2*21 + (7-5.7)^2*34}{96} = 1.3275\).
Step 4 ：Finally, calculate the standard deviation. The standard deviation is the square root of the variance. So, \(\sigma = \sqrt{Variance} = \sqrt{1.3275} = 1.1521718621802912\).
Step 5 ：Round the standard deviation to one decimal place. So, the standard deviation of the random variable X is approximately \(\boxed{1.2}\) games.
