Question

Please show your work, thank you!

Which of the following are consequences of the Central Limit Theorem? I'm not sure why II and III are correct and the others are not.

I) A SRS of resale house prices for 100 randomly selected transactions from all sale

transactions in 2001 (in Toronto) will be obtained. Since the sample is large, we

should expect the histogram for the sample to be nearly normal.

II) We will draw a SRS (simple random sample) of 100 students from all University

of Toronto students, and measure each person’s cholesterol level. The average

cholesterol level for the sample should be approximately normally distributed.

III) We want to estimate the proportion of Ontario voters who intend to vote for the

Liberal party in the next election, and decide to draw a SRS of 400 voters. The

percentage of the people in the sample who will say that they intend to vote

Liberal is approximately normally distributed.

IV) We will draw a SRS of 100 adults from the Canadian military, and count the

number who have the AIDS virus. The number of individuals in the sample who

will be found to have the AIDS virus should be approximately normally

distributed.

V) We are interested in the average income for all Canadian families for 2001. The

mean income for all Canadian families should be approximately normal, due to

the large number of families in the population.

Answer 1

Question asked: why II and III are correct and the others are not.

By Central Limit Theorem, the distribution of Sample Statistic is Normal Distribution, irrespective of the shape of the population distribution, for large samples, i.e., n > 30.

Reason for I, IV and V are not correct.

In the case of I, CLT is not applicable because: the histogram for the sample: is NOT a Sample Statistic.

In the case of IV, CLT is not applicable because: the number of individuals in the sample who will be found to have the AIDS virus is NOT a Sample Statistic.

In the case of V, CLT is not applicable because mean income for all Canadian families is the population parameter and it is NOT a sample statistic.

Reason for II and III are correct.

In the case of II, CLT is applicable because: the average cholesterol level for the sample is a Sample Statistic.

In the case of III, CLT is applicable because:the percentage of the people in the sample who will say that they intend to vote Liberal is a Sample Statistic.