central limits theorem

Question

Use central limits theorem to solve this problem and draw

conclusions from it:

A population of first year undergraduate students working part time

has a mean salary of £15,572 with a standard deviation of £3,150. If a

sample of 50 students is taken, what is the probability that the mean

of their salaries will be less than £15,000? How would this probability

change, if the sample had only 5 students?

Answer 1

Hello James! I will do my best to answer the questions above. This question is related to a normal distribution. I would normally answer this using a TI-84 calculator, so I will do that first, then discuss how to do it by hand.

First, you must take the population information given and convert it to a sampling distribution. As such, the population mean of 15,572 would also be the sample mean. μ=x-bar. Then to convert the population standard deviation to a sample standard deviation, we use the formula σ/√n . So, 3150/√50 = 445.477.

Using the TI-84 Calculator

Now, in the TI-84 calculator, you hit 2nd VARS, Option #2 (normalcdf). Since the questions asks for the probability that the mean is less than 15,000, this is a left tail question. Thus the lower limit is negative infinity which we represent at -1E99 (the E is 2nd then the comma button above the 7). The upper limit is 15000, the mean is 15572 and the standard deviation is 445.477. It will look like this: normalcdf(-1E99, 15000, 15572, 445.477). Execute that, and you will get the answer.

By Hand



To do this by hand, you will convert the 15000 into a Z-score using the formula z = (x-µ)/(σ/√n). So it will be (15000-15572)/(3150/√50) = -572/445.477 = -1.28. You will then use the Standard Normal Distribution table found in your book to look up that z-score.

Now, please let me know what answer you get after following these steps.