Question

George’s average bowling score is$180$; he bowls in a league where the average for all bowlers is $150$ and the standard deviation is $20$Bill’s average bowling score is $190$; he 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) Bill, because his standardized score is higher than George’s.

Step-by-step solution

Given information

For $1st$ bowler

$x=180,\mu =150,\sigma =20$

For $2nd$ bowler

$x=190,\mu =160,\sigma =15$

Concept

The formula used: $Z=\frac{x-\mu }{\sigma }$

Calculation

Formula for $z-$score

$$\begin{gathered} Z=\frac{x-\mu }{\sigma } \\ {Z}_{George}=\frac{180-150}{20}=1.5 \\ {Z}_{Bill}=\frac{190-160}{15}=2 \\ {Z}_{Bill}>Z_{George} \end{gathered}$$

So the correct choice is (b).