Question

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.
(c) compute and interpret the mean of the random variable x.
\\( \mu _ { x } = 5.7 \\) game(s)
(round to one decimal place as needed.)
interpret the mean of the random variable 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 many times, would be expected to last about 5.7 game(s), on average.
b. the series, if played one time, would be expected to last about game(s).

Explanation:

Step1: Calculate the total number of series

We sum up the frequencies: \(20 + 22+22 + 32=\sum f=96\)

Step2: Calculate the sum of \(x\times f\)

$$
LATEXBLOCK0
$$

Step3: Calculate the mean \(\mu_X\)

Using the formula \(\mu_X=\frac{\sum(x\times f)}{\sum f}\), we substitute the values: \(\mu_X=\frac{546}{96}\approx5.7\)

The mean of a discrete - random variable represents the long - term average value. When we say the mean of the random variable \(X\) (where \(X\) is the number of games in a series) is \(5.7\), in the context of a series of games (like a tournament series), it means that if the series (the set of games in the tournament) is played many times (over a large number of repetitions of the tournament series), on average, the number of games in the series is approximately \(5.7\). Option B is incorrect because if the series is played only once, the actual number of games will be an integer (\(4\), \(5\), \(6\) or \(7\)) and not the mean value (the mean is a long - term average).

Answer:

A. The series, if played many times, would be expected to last about \(5.7\) game(s), on average.