Question

George has an average bowling score of $180$ and bowls in a league where the average for all bowlers is $150$ and the standard deviation is $20$ Bill has an average bowling score of $190$ and bowls in a league where the average is $160$ and the standard deviation is $15$ Who ranks higher in his own league, George or Bill?

(a) Bill, because his $190$ is higher than George’s $180$

(b) Bill, because his standardized score is higher than George’s.

(c) Bill and George have the same rank in their leagues because both are $30$ pins above the mean.

(d) George, because his standardized score is higher than Bill’s.

(e) George, because the standard deviation of bowling scores is higher in his league.

Short answer

The correct option is (b).

Step-by-step solution

Step 1. Given

(a) Bill, because his score of $190$ is higher than George's score of $180$

(b) Bill, because he has a better-standardized score than George.

(c) Bill and George are in the same league since they are both $30$ pins over the mean.

(d) George, because he has a higher standardized score than Bill.

(e) George, because his league's standard deviation for bowling scores is larger.

Step 2. Concept

The z score is calculated using the following formula:$z=\frac{X-\mu }{\sigma }$

Step 3. Calculation

The summary statistics are:

$$\begin{gathered} z_{George}=\frac{x-\mu }{\sigma } \\ =\frac{180-150}{20} \\ =1.5 \end{gathered}$$

$z_{Bill}=\frac{x-\mu }{\sigma }$

$$\begin{gathered} =\frac{190-160}{15} \\ =2 \end{gathered}$$

Bill's score is $2$standard deviations higher than the mean, whereas George's is $1.5$standard deviations higher. Because Bill's standardized score is higher than George's, (b) is the correct answer. Therefore, the correct option is (b)