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

Define the standard random variable Z as $Z={X-350\over 60}$ Where X denotes the score

a) Since from the normal table we know that

$\Pr(Z\leq 1.04)=0.85$

So, Mary's score must be

${X-350\over 60}\geq 1.04 \\ \Rightarrow X\geq 412.4$

As 435> 412.4

So Mary will get admission

b) Redefine the standard rando variable Z as $Z= {\bar{X}-350\over 60/\sqrt{40}} ={\sqrt{40}(\bar{X}-350)\over 60}$ where is the sample mean.

X

Then using normal table we get

$\Pr( \bar{X} \leq 370)=\Pr\left ( Z\leq {\sqrt{40}(370-350)\over 60} \right ) =\Pr(Z\leq 2.11)=0.9826$

c) We know from normal table that $\Pr(Z\leq 1.645)=0.95$

So the sample size must be such that

${\sqrt{n}(375-350)\over 60} \geq 1.645 \\ \Rightarrow n \geq 15.58$

Therefore, the sample size required is at least 16