Let X represent the SAT total score of an entering freshman at University X.The random variable X is known to have a N(1200,90)distribution.Let Y represent the SAT total score of an entering freshman at University Y.The random variable Y is known to have a N(1215,110)distribution.A random sample of 100 freshmen is obtained from each university.Let = the sample mean of the 100 scores from University X,and = the sample mean of the 100 scores from University Y.What is the distribution of the difference in sample means between University X and University Y: - ?
A)N(-15,-20)
B)N(-15,14.2)
C)N(-15,142.1)
D)N(-15,200)

Question

Let X represent the SAT total score of an entering freshman at University X.The random variable X is known to have a N(1200,90)distribution.Let Y represent the SAT total score of an entering freshman at University Y.The random variable Y is known to have a N(1215,110)distribution.A random sample of 100 freshmen is obtained from each university.Let  = the sample mean of the 100 scores from University X,and  = the sample mean of the 100 scores from University Y.What is the distribution of the difference in sample means between University X and University Y:  -  ?
A)N(-15,-20) 
B)N(-15,14.2) 
C)N(-15,142.1) 
D)N(-15,200)

Quizplus · Accepted Answer

The difference in sample means between University X and University Y is:

11eab14b_c866_91e4_9166_6b3c27eda8ca_TB5065_11 - 11eab14b_c866_91e5_9166_fdcb8dc721a8_TB5065_11

= 11eab14b_c866_91e4_9166_6b3c27eda8ca_TB5065_11 - 11eab14b_c866_91e5_9166_fdcb8dc721a8_TB5065_11

= 1200 - 1215

= -15

The variance of the difference in sample means is equal to:

Var(11eab14b_c866_91e4_9166_6b3c27eda8ca_TB5065_11 - 11eab14b_c866_91e5_9166_fdcb8dc721a8_TB5065_11)

= Var(11eab14b_c866_91e4_9166_6b3c27eda8ca_TB5065_11) + Var(11eab14b_c866_91e5_9166_fdcb8dc721a8_TB5065_11)

= (90/100) + (110/100)

= 2

The standard deviation of the difference in sample means is equal to:

SD(11eab14b_c866_91e4_9166_6b3c27eda8ca_TB5065_11 - 11eab14b_c866_91e5_9166_fdcb8dc721a8_TB5065_11)

= sqrt(2)

≈ 1.414

Therefore, the distribution of the difference in sample means between University X and University Y is:

N(-15, 1.414^2)

≈ N(-15, 2)

which is closest to option B, N(-15, 14.2).

Let X Represent the SAT Total Score of an Entering