Question

The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean μ=542.7 and standard deviation σ=29.8.

(a) What is the probability that a single student randomly chosen from all those taking the test scores 546 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¯, of 35 students?
The mean of the sampling distribution for x¯is:  
The standard deviation of the sampling distribution for x¯ is:

(c) What z-score corresponds to the mean score x¯ of 546?
ANSWER:

(d) What is the probability that the mean score x¯ of these students is 546 or higher?
ANSWER:  

Answer 1

Solution :

Given that ,

mean = = 542.7

standard deviation = = 29.8

a) P(x > 546) = 1 - p( x< 546)

=1- p P[(x - ) / < (546 - 542.7) /29.8 ]

=1- P(z < 0.11)

Using z table,

= 1 - 0.5438

= 0.4562

b) n = 35

= = 542.7

= / n = 29.8 / 35 = 5.037

c) = 546

Using z-score formula,

z = -   /

z = 546 - 542.7 / 5.037

z = 0.66

d) P( > 546) = 1 - P( < 546)

= 1 - P[( - ) / < (546 - 542.7) / 5.037]

= 1 - P(z < 0.66)

Using z table,    

= 1 - 0.7454

= 0.2546