Question

ACT math scores for a particular year are approximately normally distributed

with a mean of 28 and a standard deviation of 2.4.
Part A: What is the probability that a randomly selected score is greater than 30.4?
Select a Value
Part B: What is the probability that a randomly selected score is less than 32.8?
Select a Value
Part C: What is the probability that a randomly selected score is between 25.6 and 32.8?
Select a Value

Answer 1

`a. \ \ \ P(X>30.4)=0.1587\\\\b. \ \ \ P(X<32.8)=0.9773\\\\c.\ \ \ P(25.6<x<32.8)=0.8186`

Explanation:

a. Given the mean is 28 and the standard deviation is 2.4

-Let X be the score of any random selection from the population.

-The probability that a randomly selected score is greater than 30.4:

`z=(\bar X-\mu)/(\sigma)\\\\P(X>30.4)=P(z>(30.4-28)/(2.4))\\\\=P(z>1)=1-P(z<1)\\\\=1-0.84134\\\\=0.1587`

Hence, the probability that the score is greater than 30.4 is 0.1587

b. The probability that a randomly selected score is less than 32.8 given mean =28 and the standard deviation is 2.4:

`z=(\bar X-\mu)/(\sigma)\\\\P(X<32.8)=P(z<(32.8-28)/(2.4))\\\\=0.9773\\\\`

Hence, the probability that a random score is less than 32.8 is 0.9773

c. The probability that a random selection has a score of between 25.6 and 32.8.

-Given the mean=28 and sd=2.4, the probability is calculated as follows:

`P(25.6<X<32.8)=P((25.6-28)/(2.4)<X<(32.8-28)/(2.4))\\\\\\=P(-1<z<2)\\\\=0.9773-0.1587\\\\=0.8186`

Hence, the probability that a random selection has a score between 25.6 and 32.8 is 0.8186