Question

World series. The World series in baseball is won by the first team to win four games ( ignoring the 1903 and 19919 - 1921 World series, when it was a best of nine). From the document World series history on the baseball Almanac website. as of November 2013, the length of the world series are as given in the following table.

Let X denote the number of games that it take to complete a world series, and let Y denote the number of games that it took to complete a randomly selected world series from among those considered in the table.

Part (a) Determine the mean and standard deviation of the random variable Y. Interpret your resuts.

Part (b) Provide an estimate for the mean and standard deviation of the random variable X. Explain your reasoning

Short answer

Part (a) The mean value 7

Part (b)

5.72 is the projected mean value.

1.122 is the estimated standard deviation.

Step-by-step solution

Part (a) Step 1. Given information. 

Consider the following table of data. The number of games required to complete the randomly selected world series is the random variable Y.

Part (a) Step 2. Interpret the random variable Y's mean and standard deviation

Mean value

$$\begin{gathered} \mu =\sum _{}y·P(Y=y) \\ \mu =4\left(0.2\right)+5(0.229)+6\left(0.229\right)+7\left(0.343\right) \\ \mu =0.8+1.15+1.4+2.4 \\ \mu =5.72 \end{gathered}$$

Standard deviation

$$\begin{gathered} \sigma =\sqrt{\sum _{}{x}^{2}P(X=x)-{\mu }^2} \\ \sigma =\sqrt{16\left(0.2\right)+25\left(0.229\right)+36\left(0.229\right)+49\left(0.343\right)-{\left(5.72\right)}^2} \\ \sigma =\sqrt{1.2576} \\ \sigma =1.1214 \end{gathered}$$

The average number of games needed to finish the world series = 5.72

The standard deviation = 1.122

Interpretation

The average number of games needed to complete a world series is 1.122, compared to a mean of 5.72.

Part (b) Step 1. Given information. 

Consider the following table of data. The number of games required to complete the world series is the random variable X.

Part (b) Step 2.  The X and Y random variable estimates

The number of games required to complete the randomly selected world series is the random variable Y. The random variable X represents the number of games needed to complete the world series.

For already completed games, the random variable Y is defined.

As a result, the random variable X has the same estimate as the random variable Y.

As a result, the mean is 5.72.

1.122 is the standard deviation.