Question

A psychology professor assigns letter grades on a test according to the following scheme. A: Top 7% of scores B: Scores below the top 7% and above the bottom 64% C: Scores below the top 36% and above the bottom 25% D: Scores below the top 75% and above the bottom 6% F: Bottom 6% of scores Scores on the test are normally distributed with a mean of 78.4 and a standard deviation of 7.6. Find the minimum score required for an A grade. Round your answer to the nearest whole number, if necessary.

Answer 1

Answer:

The minimum score required for an A grade is 89.8.

Step-by-step explanation:

We are given that a psychology professor assigns letter grades on a test according to the following scheme. A : Top 7% of scores. B : Scores below the top 7% and above the bottom 64%. C : Scores below the top 36% and above the bottom 25%. D : Scores below the top 75% and above the bottom 6%. F : Bottom 6% of scores.

Scores on the test are normally distributed with a mean of 78.4 and a standard deviation of 7.6.

Let X = Scores on the test

SO, X ~ Normal($\mu=78.4,\sigma^{2} =7.6^{2}$)

The z-score probability distribution for normal distribution is given by;

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

where, $\mu$ = mean time = 78.4

            $\sigma$ = standard deviation = 7.6

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X.

Now, the minimum score required for an A grade so that it represents Top 7% of scores is given by;

      P(X $\geq$ x) = 0.07   {where x is the required minimum score

      P( $\frac{X-\mu}{\sigma}$ $\geq$ $\frac{x-78.4}{7.6}$ ) = 0.07

       P(Z $\geq$ $\frac{x-78.4}{7.6}$ ) = 0.07

So, the critical value of x in the z table which represents the top 7% of the area is given as 1.4996, that is;

                        $\frac{x-78.4}{7.6} =1.4996$

                        ${x-78.4}{} =1.4996\times 7.6$

                        $x$  = 78.4 + 11.39696 = 89.8 or 90

Hence, the minimum score required for an A grade is 89.8.