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) (79) 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

a)

Here,
T 11 o
, where, $\mu=350$ , and,
09 = 0

Here,

C = 435

Therefore, ,

435-350 60

i.e., $z=1.4166$

$P(X>x)=0.07829$, and,

P(X <C) = 0.92171

That is, 7.829 % of students, score more than Mary, and, 92.17 % of students score less than Mary.

Therefore we can conclude that Mary will be admitted to that certain university.

b)

Here,
T 11 o
, where, $\mu=350$ , and,
09 = 0

Here, $x=375$

Therefore, ,

370 — 350 2 60

i.e., $z=0.3333$

$P(X<x)=0$,

Therefore, If a random sample of 40 students is selected, the probability that the sample average of test scores is at most 370, is, 0.

c)

Here,
T 11 o
, where, $\mu=350$ , and,
09 = 0

Here, $x=375$

Therefore, $z=\frac{375-350}{60}$ ,

i.e., $z=0.4166$

$P(X \ge x)=1-P(X \le x)$,

Here, $P(X \ge x)=0.95$

Therefore, $0.95=1-P(X \le x)$

i.e., $P(X \le x)=0.05$

The required sample size=(0.05)(375)=18.75

i.e., The required sample size=(0.05)(375)=19