Qualitative Stats question (central limit theorem)

In summary, the Central Limit Theorem states that for large sample sizes, the distribution of sample means will be approximately normal. This applies to various scenarios, such as measuring cholesterol levels of students at a university, estimating the proportion of voters for a political party, and counting the number of individuals with a certain condition in a sample. However, for the approximation to be useful, the sample size and proportion must be large enough. Otherwise, the normal approximation will be poor and not provide accurate results.
  • #1
habman_6
16
0
3) Which of the following are consequences of the Central Limit Theorem?
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.


The answer is II and III. I understand why I is wrong, and I understand why V is wrong. However, to me, IV seems exactly the same as III. Apparently its because the proportion is too low, but that does not make sense to me in terms of CLT. What should it matter what the proportion is, as long as the sample means have that same proportion?
 
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  • #2
If the sample size or the proportion is too small the approximation is too poor to be useful. As an example, consider the Binomial distribution(N,p). The CLT applies to this case. The rule of thumb is that the normal approximation to the Binomial is poor unless Np > 10.

http://www.stat.yale.edu/Courses/1997-98/101/binom.htm

To convince yourself, look at the pdf of some binomial distributions that don't satisfy Np > 10, e.g. N = 100, p = 0.01.
 
  • #3
Thanks Aleph, I understand now.
 

Related to Qualitative Stats question (central limit theorem)

1. What is the central limit theorem?

The central limit theorem is a fundamental concept in statistics that states that as sample size increases, the distribution of sample means will approach a normal distribution, regardless of the shape of the population distribution.

2. How is the central limit theorem used in qualitative statistics?

In qualitative statistics, the central limit theorem is used to determine the distribution of sample means for a given population. This allows researchers to make inferences about the population based on the sample data.

3. What are the assumptions of the central limit theorem?

The central limit theorem assumes that the sample is randomly selected from the population, the sample size is large enough, and the observations within the sample are independent. These assumptions are necessary for the theorem to hold true.

4. Can the central limit theorem be applied to non-normal populations?

Yes, the central limit theorem can be applied to non-normal populations. As long as the sample size is large enough, the distribution of sample means will approach a normal distribution, even if the population itself is not normally distributed.

5. What is the practical significance of the central limit theorem?

The central limit theorem is important in statistics because it allows researchers to make confident inferences about a population using sample data. It also allows for the use of parametric statistical tests, which rely on the assumption of normality, even if the population is not normally distributed.

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