Question

3. A division wide aptitude test in Mathematics was conducted to 3 800 students. The mean of the test is 58 and the standard deviation is 14.

a. What is the score that divides the distribution into two such that 80% of the cases is below it.

b. What are the scores that bounds the middle 90% of the distribution.

c. How many scores are between 60 and 96?

d. How many students belong to the top 12% of the examinees?

Answer 1

Let $X=$ score, $X\sim N(\mu, \sigma^2).$

a.

$$P(X<x)=P(Z<\dfrac{x-\mu}{\sigma})$$

$$=P(Z<\dfrac{x-58}{14})=0.8$$

$$\dfrac{x-58}{14}\approx0.841621$$

$$x=69.78$$

b.

$$P(X<x_1)=P(Z<\dfrac{x_1-\mu}{\sigma})$$

$$=P(Z<\dfrac{x_1-58}{14})=0.05$$

$$\dfrac{x_1-58}{14}\approx-1.6449$$

$$x_1=35$$

$$P(X>x_2)=P(Z>\dfrac{x_2-\mu}{\sigma})$$

$$=P(Z>\dfrac{x_2-58}{14})=0.05$$

$$\dfrac{x_2-58}{14}\approx1.6449$$

$$x_2=81$$

Between 35 and 81.

c.

$$P(60<X<96)$$

$$=P(Z<\dfrac{96-58}{14})-P(Z\le\dfrac{60-58}{14})$$

$$\approx0.99668-0.55680\approx0.43988$$

$$0.43988(3800)=1672$$

1672 scores are between 60 and 96.

d.

$$0.12(3800)=456$$

456 students belong to the top 12% of the examinees.