Question

the scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean μ=536.1μ=536.1 and standard deviation σ=27.9σ=27.9.

(a) What is the probability that a single student randomly chosen from all those taking the test scores 540 or higher?
ANSWER:  

For parts (b) through (d), consider a simple random sample (SRS) of 35 students who took the test.

(b) What are the mean and standard deviation of the sample mean score x¯x¯, of 35 students?
The mean of the sampling distribution for x¯x¯ is:  
The standard deviation of the sampling distribution for x¯x¯ is:

(c) What z-score corresponds to the mean score x¯x¯ of 540?
ANSWER:

(d) What is the probability that the mean score x¯x¯ of these students is 540 or higher?
ANSWER:  

Answer 1

Solution :

Given that ,

mean = = 536.1

standard deviation = = 27.9

a) P(x > 540) = 1 - p( x< 540)

=1- p P[(x - ) / < (540 - 536.1) / 27.9]

=1- P(z < 0.14)

Using z table,

= 1 - 0.5557

= 0.4443

b) n = 35

= = 536.1

= / n = 27.9 / 35 = 4.72

c) z = -   /

z = 540 - 536.1 / 4.72

z = 0.83

d) P ( Z > 0.83)

= 1 - P(z < 0.83)

Using z table,    

= 1 - 0.7967

= 0.2033