Chapter 12.5, Problem 3CYP

To determine

To find: The number of students having a grade-point average between $2.0$ and $3.2$ .

Answer to Problem 3CYP

The number of students having a grade-point average between $2.0$ and $3.2$ is $819$ .

Explanation of Solution

Given information:

Total number of students $=1200$

Mean of grade-point averages $=2.6$

Mean of grade-point averages $=0.6$

Calculation:

Let us define the random variable ,

  $X$ : Grade point average

Given that $X$ follows Normal distribution with mean, $\mu =2.6$ and standard deviation, $\sigma =0.6$ .

  $Z$ score of $X$ is given by

  $Z=$ ( $X-$ Mean of $X$ ) / Standard deviation of $X$

  $=\frac{X-\mu }{\sigma }$

  $Z$ score of $3.2$ is $\frac{3.2-2.6}{0.6}=1$

  $Z$ score of $2$ is $\frac{2-2.6}{0.6}=-1$

The probability that a randomly selected student has grade point average between $2.0$ and $3.2$ is $P(2.0<X<3.2)=P(X<3.2)-P(X<2.0)$

  $=P(Z<1)-P(Z<-1)$

  $=0.8413-0.1587$ ( from $Z$ table )

  $=0.6826$

Out of $1200$ students, the number of students who have grade −point average between $2.0$ and $3.2$ is $1200\times P(2.0<X<3.2)=1200\times 0.6826$

  $=819.12$

  $=819$ ( rounded to nearest integer)

Thus, the number of students having a grade-point average between $2.0$ and $3.2$ is $819$ .