Business Week conducted a survey of graduates from 30 top MBA programs (Business Week, September $22,2003$ . On the basis of the survey, assume that the mean annual salary for male and female graduates 10 years after graduation is $\$ 168,000$ and $\$ 117,000$ , respectively. Assume the standard deviation for the male graduates is $\$ 40,000,$ and for the female graduates it is $\$ 25,000$ .
a. What is the probability that a simple random sample of 40 male graduates will provide a sample mean within $\$ 10,000$ of the population mean, $\$ 168,000 ?$
b. What is the probability that a simple random sample of 40 female graduates will provide a sample mean within $\$ 10,000$ of the population mean, $\$ 117,000$ ?
c. In which of the preceding two cases, part (a) or part (b), do we have a higher probability of obtaining a sample estimate within $\$ 10,000$ of the population mean? Why?
d. What is the probability that a simple random sample of 100 male graduates will provide a sample mean more than $\$ 4000$ below the population mean?
c. Part (b) is higher, because the standard deviation for males is higher than for females.
all right, were given some data about graduates and there salaries 10 years after graduation, and we've divided them into data for men and data from women. So for part A, given the statistics Ah, we're given a sample of 40 men, and we want to find the probability that our sample mean will be within 10,000 of our population. Means hold on. Gonna crack that notation. There we go on that might need his penmanship, but you get the general idea, right? So let's find our standard deviation of the sampling distribution. It's gonna be the standard deviation over sample size. So that's gonna be, uh, 40,000 divided by the square root of 40. That equals 6324 56 We're gonna find a Z lower and see upper like so. So for a C lower, uh, it's gonna be negative. 10,000 divided by our center deviation, uh, for the sample. Yeah, our sampling distribution, and then for upper, it's gonna be 10,000 positive. When you compute these out, you get negative 1.58 from 1.58 Comparing this to our normal probabilities table. This gives you probability. Lower 0.571 probability, upper 0.9429 Uh, this means our probability is gonna be probability upper minus probability, lower. Which is zero point 8858 Right party. Uh, this time we're looking at a sample of 40 women, and now we need to find the probability that our sample mean that we find is within 10,000 of the mean for the women. So once again, we're going to find our standard deviation sampling distribution. That's gonna be this time. We're looking at the statistics for the women. So this is going to be 25,000 over square root of 40. Calculate that out. That's 3952.47 are Sorry. 847 Anyway, let's find a Z lowers the upper. So the els you again. That's gonna be negative. 10,000 over our standard deviation of the sampling distribution for women. This one's gonna be 10,000 positive. So these are equal negative. 2.53 2.53 respectively. This is P. L. Looking at our table Once again, this lower probability is from zero point 0057 Upper probability 0.9943 Finding a probability, which is probability Oper minus probability, lower. You get, um, zero point 9886 part C s us two compared these to given explanation why one is higher than the other. And we see that part. He is greater. This is because standard deviation for men is greater then that for women I'm gonna have to move that up on. And because the standard deviation is greater, this means that this is a small This is smaller relative to our standard deviation, which means this let fewer standard deviations away from the mean, as opposed to this. All right, I'm gonna move to a different page party. Now we have sample of 100 men. We want to find the probability that air sample mean is within. Ah, I believe it is sorry. It's not within this time. We need to find a sample mean that is greater than our population mean minus 4000. All right, so let's find our sampling distribution. Ah, standard deviation. So that's referring back to hear. That's 40,000 square root of 100 square. 100 is tense. And this just 4000. All right, now we just see score for 4000 against 4000. So this could be because we are looking at 4000 less than the mean That's gonna be negative. 4000 top standard deviation is 4000. So this is negative 1.0 And if we look at our table, this corresponds to a probability is zero point 1587 and there you have it.