Question

A distribution of exam scores has mean $60$and standard deviation $18$. If each score is doubled, and then $5$is subtracted from that result, what will be the mean and standard deviation, respectively, of the new scores?

(a) mean$=115$and standard deviation$=31$

(b) mean$=115$and standard deviation$=36$

(c) mean$=120$and standard deviation$=6$

(d) mean$=120$and standard deviation$=31$

(e) mean$=120$ and standard deviation$=36$

Short answer

The answer is mean$115$and standard deviation $36$. so option (b) is correct.

Step-by-step solution

Given Information

We are given that the mean is ${\mu }_x=60$and standard deviation is ${\sigma }_x=18$and we have to find out mean and standard deviation when $5$is subtracted from the result.

Explanation

Now as we have old mean and standard deviation,

which gives 60 and 18 respectively,

But there is a property of mean and standard deviation,

${\mu }_{ax+b}=a{\mu }_x+b$(property of mean)

${\sigma }_{ax+b}=a{\sigma }_x$(property of standard deviation)

Now $ax+b=2x-5$

and putting this in the properties,

we get,${\mu }_{2x-5}=2{\mu }_x-5=2\left(60\right)-5$

On solving we get $newmean=115$

and also, ${\sigma }_{2x-5}=2{\sigma }_x=2\left(18\right)$

On solving we get, $news\tan darddeviation=36$

Hence, option (b) is correct.