Answer to Question #332071 in Statistics and Probability for Jefftzy

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.