Question

PLEASE SOLVE ALL PARTS AND EXPLAIN IN DETAIL.

I'M HAVING A DIFFICULT TIME UNDERSTANDING HOW TO FIND/SOLVE THESE TYPES OF PROBLEMS:

In 2017, as reported by ACT Research Service, the mean ACT Math score was μ=20.6 μ=20.6 . ACT Math scores are normally distributed with σ=4.8 σ=4.8 .

1. What is the probability that a randomly selected test taker has an ACT Math score less than 18? Is this result unusual? Explain.
2. Describe the sampling distribution of x ˆ  x^ , the sample mean ACT Math score for a random sample of 40 tests using the N(mean, standard deviation) notation.
3. What is the probability that a random sample of 40 tests has a mean ACT Math score less than 18? Is this result unusual? Explain.
4. What is the difference in part a and c in this question?

Answer 1

a)

µ =    20.6          
σ =    4.8          
              
P( X ≤    18   ) = P( (X-µ)/σ ≤ (18-20.6) /4.8)      
=P(Z ≤   -0.542   ) =   0.2940   (answer)

this is not unusual because probability is greater than 0.05

b)

mean=20.6

std dev = σ/√n = 4.8/√40 = 0.7589

N(20.6,0.7589)

c)

µ =    20.6                                      
σ =    4.8                                      
n=   40                                      
                                          
X =   18                                      
                                          
Z =   (X - µ )/(σ/√n) = (   18   -   20.60   ) / (   4.800   / √   40   ) =   -3.43  
                                          
P(X ≤   18   ) = P(Z ≤   -3.426   ) =   0.0003                       (answer)

this is unusual because probability is less  than 0.05

d)

in part a) , probability is calculated for a random test taker.

in part c), probability is calculated for average of 40 random sample test taker