Answer to Question #339831 in Statistics and Probability for felicity

Question

2.) Scores on the SAT form a normal distribution with a mean score of 500 and a standard deviation of 100.

a. What is the minimum score necessary to be in the top 15% of the SAT distribution?

b. Find the range of scores that defines the middle 80% of the distribution of SAT scores.

3. )The government would like to conduct a subsidy program for the lowest 5 percent of the families in terms of income. The government gathered data about family income and it’s found to be normally distributed with a mean of Php 130 000 and a standard deviation of Php 50 000. What is the cutoff income for the government program?

Answer 1

2.)

a)

$$P(X\le x)=1-0.15$$

$$P(Z\le \dfrac{x-500}{100})=0.85$$

$$\dfrac{x-500}{100}=1.036433$$

$$x=603.64$$

Minimum score is 603.64

b)

$$P(X<x_1)=0.1$$

$$P(Z< \dfrac{x_1-500}{100})=0.1$$

$$\dfrac{x_1-500}{100}=-1.28155$$

$$x_1=371.845$$

$$\dfrac{x_2-500}{100}=1.28155$$

$$x_2=628.155$$

$$P(371.845\le X\le 628.155)=0.8$$

3.)

$$P(X<x)=0.05$$

$$P(Z< \dfrac{x-130000}{50000})=0.05$$

$$\dfrac{x-130000}{50000}=-1.6449$$

$$x=47755$$