Question

Suppose SAT Writing scores are normally distributed with a mean of 489489 and a standard deviation of 112112. 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 whole number, if necessary.

Answer 1

Answer:

The minimum score required for the scholarship is 665.09.

Step-by-step explanation:

We are given that SAT Writing scores are normally distributed with a mean of 489 and a standard deviation of 112.

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

Let X = SAT writing scores

SO, X ~ N($\mu = 489,\sigma^{2} = 112^{2}$)

The z-score probability distribution is given by ;

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

where, $\mu$ = mean score = 489

            $\sigma$ = standard deviation = 112

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 the minimum score required}

             P( $\frac{X-\mu}{\sigma}$ $\geq \frac{x-489}{112}$ ) = 0.06

             P(Z $\geq \frac{x-489}{112}$ ) = 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;         $\frac{x-489}{112} = 1.57224$

                          $x-489=1.57224 \times 112$  

                              $x$ = 489 + 176.0909 = 665.09

Therefore, the minimum score required for the scholarship is 665.09.