Question

Suppose ACT Reading scores are normally distributed with a mean of 21.221.2 and a standard deviation of 6.26.2. A university plans to award scholarships to students whose scores are in the top 6%6%. What is the minimum score required for the scholarship? Round your answer to the nearest tenth, if necessary.

Answer 1

Answer:

The minimum score required for the scholarship is 30.95.

Step-by-step explanation:

We are given that ACT Reading scores are normally distributed with a mean of 21.2 and a standard deviation of 6.2.

A university plans to award scholarships to students whose scores are in the top 6%.

Let X = ACT Reading scores

SO, X ~ N($\mu = 21.2,\sigma^{2} = 6.2^{2}$)

The z-score probability distribution is given by ;

                  Z = $\frac{X-\mu}{\sigma}$ ~ N(0,1)

where, $\mu$ = mean score = 21.2

            $\sigma$ = standard deviation = 6.2

Now, the minimum score required for the scholarship so that students  are in the top 6% is given by ;

              P(X $\geq$ $x$ ) = 0.06   {where $x$ is minimum score required}

             P( $\frac{X-\mu}{\sigma}$ $\geq \frac{x-21.2}{6.2}$ ) = 0.06

             P(Z $\geq \frac{x-21.2}{6.2}$ ) = 0.06

Now, in z table we will find out that critical value of X for which the area is in top 6%, which comes out to be 1.57224.

This means;         $\frac{x-21.2}{6.2} = 1.57224$

                          $x-21.2=1.57224 \times 6.2$  

                              $x$ = 21.2 + 9.7478 = 30.95

Therefore, the minimum score required for the scholarship is 30.95.