Question

1.) In a National Achievement Test, the mean was found to be 75 and the standard deviation was 15. The scores also approximate the normal distribution.

a. What is the minimum score that belongs to the upper 15% of the group?

b. What is the two extreme scores outside of which 15% of the group are expected to fall?

c. What is the score that divide the distribution into two such that 75% of the cases below it?

d. Estimate the range of scores that will include the middle 45% of the distribution.

Answer 1

a.

$$P(X>x)=1-P(Z\le \dfrac{x-75}{15})=0.15$$

$$\dfrac{x-75}{15}=1.036433$$

$$x=90.5465$$

b.

$$P(X<x_1)=\dfrac{0.15}{2}$$

$$P(Z\le \dfrac{x_1-75}{15})=0.075$$

$$\dfrac{x_1-75}{15}=-1.4395$$

$$x_1=53.4075$$

$$P(X>x_2)=\dfrac{0.15}{2}$$

$$1-P(Z\le \dfrac{x_2-75}{15})=0.075$$

$$\dfrac{x_2-75}{15}=1.4395$$

$$x_2=96.5925$$

c.

$$P(X<x)=P(Z< \dfrac{x-75}{15})=0.75$$

$$\dfrac{x-75}{15}=0.6749$$

$$x=85.1235$$

d.

Middle 45% is contained in the range of 27.5%-72.5%

$$P(X<x_1)=P(Z< \dfrac{x_1-75}{15})=0.275$$

$$\dfrac{x_1-75}{15}=-0.59776$$

$$x_1=66.0336$$

$$P(X<x_2)=P(Z< \dfrac{x_2-75}{15})=0.725$$

$$\dfrac{x_2-75}{15}=0.59776$$

$$x_2=83.9664$$