Question

5- A national test is held to admit students to the public universities. The scores on this test are normally distributed with a mean of 350 and a standard deviation of 60. Mary knows if she wants to be admitted to a certain university, her score should be better than at least 85% of the students who took the test.

a) (6%) Mary takes the test and scores 435. Will she be admitted to that certain university?

b) (7%) If a random sample of 40 students selected, what is the probability that the sample average of test scores is at most 370?

c) (7%) How large a sample size would be required to ensure that, the probability of having the average of 375 for test scores is at least 0.95?

Answer 1

let the scores are random variable x XnNl350 (60)²] we know that it from NCM.02) then. Z = X - M رله ما P(XK 435)=P(x-350 435-350 60 P(2.4 1.416666) from 2-table calculator P(x < 435) - 0.92171 a 92% So She will get the admission as she is better than the students. 92% y 6 A180 et on N(H₂O2) then 7 ~ N(M, 02) where n is sample size n = 90 n XNN (350, thony P(x4370)=P(x-350 370-350 607850 = P(Z < 2,108185) From z-table calculator. P(<370) = 0.98249

n = ? P(x<375) - 0.95 P(x-350 375-350 60/ h ) = 0.95 Golsh plek 375-350 -0.95 60/8n corresponding to from 2-table calculator the 2-score 0.95 is -1.645 So 375-350 1.645 60/55 1.645 x 60 25 un = 3.948 n 15.5867 So sample size of 16 is required