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?

Answer 1

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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?

So for scores we know, with mean =$\mu$ = 350 and std. dev = $\sigma$ =60

a) Mary Scores 435. This score 435 will represent a point (z-$\mu$)/$\sigma$ = (450-350)/60 = 1.666

(z+$\mu$)/$\sigma$ = 800/60 = 13.333

So we have our normal distribution curve as follows

4-30 *4 2-- Z=0

The shaded area will represent the % of students who are better than her.

From normal distribution tables we find this to be equal to 1 - 0.9515 = 0.0485

Table of Values for N(2) N(Z) = shaded area For Negative Values: N(-2) = 1 - N(Z) 0 z how.com T е n t h S Hundredths Digits 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359 0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753 0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141 0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517 0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879 0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224 0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549 0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852 0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133 0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389 1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621 1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830 1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015 1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177 1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319 1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441 1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505.9515 0.9525 0.9535 0.9545 1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633 1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706 1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767 D 8 i t s בדרום הר 9779 0792 09090702 0917

So Mary's score is better than 95.15% of the students who took the test. So yes MARY WILL BE ADMITTED TO THE UNIVERSITY

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b) We want probability of test scores to be 370. So again mark up this point on the normal distribution curve.

A score of 370 is represented by a point (z-$\mu$)/$\sigma$ = (370-350)/60 = 0.333 on the normal distribution curve.

件30 350 7-0333 2-

So probability of scores being atlmost 370 = 1 - shaded area.

From Z tables probability of shaded area = 1 - Probabillity value of Z being .333 = 1 - .6293 = 0.3707

So probability of scores being at 370 = .3707

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c) Probability of scores bein atleast 375 is represented by a point of (375-350)/60 = .4166 on the normal distribution curve.

So from z tables, probability of scores being at least 375 is given by .6628

For normal distribution,  probability is solely dependent on mean and standard deviation. Thus the mean of the distribution of the means never changes.

So no matter how large your sample size is the mean will remain at 350.

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