Can someone please help me with my statistics homework!!?

Question

in past years, males have earned higher mean scores in he verbal and math sections on the SAT. on the new writing section, male scores followed a N (491, 110) distribution. Female scores followed a N (502, 108) distribution.

a) what percentage of male test takers scores below the mean score for female test takers?

b) what percentage of female test takers earned scores above the mean score for male test takers?

c) what percentage of male test takers earned scored above the 85th percentile of the female test takers?

please help!!!

Answer 1

P(X < 502.0) = P((x - 491) / 10.5) < (502.0 - 491) / 10.5) = P(Z < 1.05) = 0.8529

P(X < 491.0) = P((x - 502) / 10.4) < (491.0 - 502) / 10.4) = P(Z < -1.06) = 0.1449

P(X > 491.0) = 1 - 0.1449 = 0.8551

P(X < 512.8) = P((x - 491) / 10.5) < (512.8 - 491) / 10.5) = P(Z < 2.08) = 0.9810

P(X > 512.8) = 1 - 0.9810 = 0.0190

Answer 2

a) The mean for the female test takers is 502 - 491 = 11 points above the mean for the male test takers. As 110 points represents one standard deviation for the male test takers, we have that the mean for the female test takers is 11/110 = .1 standard deviation above the mean of the male test takers. (In the terminology of statistics, z = .1.) So the question becomes, What percentage of the male test takers scored less than .1 standard deviation above the male mean? By using whatever statistical table your text has, the answer in decimal form is .5398 and in percentage is 53.98%.

b) The mean for the female test takers is 502 - 491 = 11 points above the mean for the male test takers. As 108 points represents one standard deviation for the female test takers, we have that the mean for the male test takers is 11/108 = .10185 standard deviation below the mean of the female test takers. So the question becomes, What percentage of the female test takers scored more than .10185 standard deviation below the female mean? By using whatever statistical table your text has, the answer in decimal form is approximately.541 and in percentage is 54.1%.

c) The 85th percentile of the female test takers corresponds to approximately 1.4 standard deviations above the female mean (as you can verify from your text's statistical table). As there are 108 points per standard deviation for the female test takers, a female test taker scoring at the 85th percentile has an SAT score of 502 + (108)(1.4) ≈ 653. So the question becomes, What percentage of the male test takers scored above 653? Now 653 points is (653 - 491)/110 ≈ 1.47 standard deviations above the male mean. From the statistical table of your text, x% of the male test takers scored below 1.47 standard deviations above the male mean, and consequently (100-x)% of the male test takers scored above 1.47 standard deviations above the male mean.

I noticed that I made an error in part c) (the 85th percentile likely corresponds to 1.40 standard deviations or maybe 1.44 standard deviations, it certainly does not correspond to 1.04 standard deviations, as you may well have noticed from your text's table). As I do not have a normal distribution table immediately available, I can't be certain about a more precise answer to part c). For the error, I apologize.

I can say that it must be in the vicinity of 12%.