Question

assume that in 2018 the mean mathematics sat score was 536 and the standard deviation was 115. a sample of 68 scores is chosen. a) what is the sampling distribution of *? b) what is the probability the sample mean score is less than 510? c) what is the probability the sample mean score is between 485 and 525? d) what is the probability the sample mean score is greater than 480? e) would it be unusual for the sample mean to be greater than 575? show work to prove your answer.

Answer 1

Solution :

Given that ,

mean = = 536

standard deviation = = 115

a) n = 68

=   = 536

= / n = 115/ 68 = 13.95

b) P( < 510) = P(( - ) / < (510 - 536) / 13.95)

= P(z < -1.86)

Using z table

= 0.0314

c) P(485 < < 525)  

= P[(485 - 536) /13.95 < ( - ) / < (525 - 536) / 13.95)]

= P(-3.66 < Z < -0.79)

= P(Z < -0.79) - P(Z < -3.66)

Using z table,  

= 0.2148 - 0.0001  

= 0.2147

d) P( > 480) = 1 - P( < 480)

= 1 - P[( - ) / < (480 - 536) / 13.95]

= 1 - P(z < -4.01)

Using z table,    

= 1 - 0

= 1

e) P( > 575) = 1 - P( < 575)

= 1 - P[( - ) / < (575 - 536) / 13.95]

= 1 - P(z < 2.80)

Using z table,    

= 1 - 0.9974

= 0.0026

Yes, it would be unusual for the sample mean to be greater than 575, because probability less than 5%