The scores of students on the SAT college entrance examinati

The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean \\mu = 544.7 and standard deviation \\sigma = 27.9.

(a) What is the probability that a single student randomly chosen from all those taking the test scores 551 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 \\bar x, of 35 students? The mean of the sampling distribution for \\bar x is: The standard deviation of the sampling distribution for \\bar x is:

(c) What z-score corresponds to the mean score \\bar x of 551? ANSWER:

(d) What is the probability that the mean score \\bar x of these students is 551 or higher? ANSWER:

Solution

Mean ( u ) =544.7
Standard Deviation ( sd )=27.9
Normal Distribution = Z= X- u / sd ~ N(0,1)                  
a)
P(X < 551) = (551-544.7)/27.9
= 6.3/27.9= 0.2258
= P ( Z <0.2258) From Standard Normal Table
= 0.5893                  
P(X > = 551) = (1 - P(X < 551)
= 1 - 0.5893 = 0.4107                  

b)
Number ( n ) = 35
Standard Deviation ( sd )=27.9/ Sqrt ( 35 ) = 4.716

c)
P(X = 551) = (551-544.7)/27.9/ Sqrt ( 35 )
Z = 6.3/4.716= 1.3359
d)
P(X < 551) = (551-544.7)/27.9/ Sqrt ( 35 )
= 6.3/4.716= 1.3359
= P ( Z <1.3359) From Standard NOrmal Table
= 0.9092                  
P(X > = 551) = 1 - P(X < 551)
= 1 - 0.9092 = 0.0908