Question

Suppose X is normally distributed with a mean of 100 and a variance of 121. Suppose a sample of size 70 is collected and the sample mean calculated. Answer the following. Enter your final answers in the boxes below. Show ALL working by uploading your handwritten working for this question to Laulima Dropbox within 15 minutes of completing your exam. Correct answers without working shown will not receive full credit. Note that writing Excel functions does NOT qualify as showing working. a) (3) Calculate the probability the sample mean is less than 103. In your working, draw a diagram showing the probability you are calculating. b) (2) Would your methodology from (a) be valid if X instead followed a Binomial distribution? Explain why or why not.

Answer 1

a) We are given the underlying distribution here as:

a) For a sample size of n = 70, using the Central limit theorem, we obtain the distribution of sample mean here as:

The probability that the sample mean is less than 103 is computed here as:

Converting it to a standard normal variable, we get here:

Getting it from the standard normal tables, we get here:

Therefore 0.9887 is the required probability here.

This could be put in a graph form as:

b) Yes, even if the underlying distribution was a binomial distribution, the above methodology would have been valid. This is because Central limit theorem holds for any underlying distribution for a sample size of greater than or equal to 30. As 70 > 30 here, therefore yes above methodology would still have been valid here.