Questions: The following data represent the number of games played in each series of an annual tournament from 1923 to 2018. Complete parts (a) through (d) below. x (games played) 4 5 6 7 Frequency 18 22 22 33 (c) Compute and interpret the mean of the random variable X. μX= □ 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 one time, would be expected to last about □ game(s).

Solution Steps

To compute the mean of the random variable \( X \), we use the formula:

\[ \mu_X = \sum (x_i \cdot P(x_i)) \]

where \( x_i \) represents the number of games played and \( P(x_i) \) is the corresponding probability. The calculations are as follows:

\[ \mu_X = 4 \times 0.1895 + 5 \times 0.2316 + 6 \times 0.2316 + 7 \times 0.3474 = 5.7 \]

The variance \( \sigma^2 \) is calculated using the formula:

\[ \sigma^2 = \sum ((x_i - \mu_X)^2 \cdot P(x_i)) \]

Substituting the values, we have:

\[ \sigma^2 = (4 - 5.7)^2 \times 0.1895 + (5 - 5.7)^2 \times 0.2316 + (6 - 5.7)^2 \times 0.2316 + (7 - 5.7)^2 \times 0.3474 = 1.3 \]

The standard deviation \( \sigma \) is the square root of the variance:

\[ \sigma = \sqrt{\sigma^2} = \sqrt{1.3} \approx 1.1 \]

The mean of the random variable \( X \) indicates that the series, if played one time, would be expected to last about \( 5.7 \) game(s).

Final Answer