Question

The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean μ=531.5μ=531.5 and standard deviation σ=29.6σ=29.6.

(a) What is the probability that a single student randomly chosen from all those taking the test scores 536 or higher?
ANSWER:  

For parts (b) through (d), consider a random sample of 25 students who took the test.

(b) What are the mean and standard deviation of the sample mean score x¯x¯, of 25 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 536?
ANSWER:

(d) What is the probability that the mean score x¯x¯ of these students is 536 or higher?
ANSWER:  

Answer 1

Solution :

Given that ,

mean = = 531.5

standard deviation = = 29.6

(a)

P(x 536) = 1 - P(x   536)

= 1 - P((x - ) / (536 - 531.5) / 29.6)

= 1 -  P(z 0.1521)  

= 1 - 0.5604

= 0.4396

P(x 536) = 0.4396

Probability = 0.4396

(b)

n = 25

The sampling distribution of mean and standard deviation is ,

= 531.5 and

= / n = 29.6 / 25 = 5.92

(c)

z = (( - ) / = (536 - 531.5) / 5.92 = 0.76

z - score = 0.76

(d)

P( > 536) = 1 - P( < 536)

= 1 - P(( - ) / < (536 - 531.5) / 5.92)

= 1 - P(z < 0.76)

= 1 - 0.7764

    Using standard normal table,

P( >536) = 0.2236

Probability = 0.2236