Question

(a)Does the Central Limit Theorem indicate that all samples follow a normal distribution if the sample size is large enough? Explain.

(b)Suppose that the observational units in a study are people in your home state, and the variable of interest in a study is number of siblings. If the sample size is chosen to be in the thousands, would a histogram of the sample data follow a normal distribution? Explain.

(c) Suggest a statistic for which your class would give biased results, that is, for which your Math 124 class would not be considered a representative sample of all a university students. Does your statistic over or under estimate the population parameter? Explain your choice.

(d)Suggest a statistic for which it might be reasonable to consider your Math 124 class giving unbiased results, that is, for which your Math 124 class could be considered representative of all students at a university. Justify your choice.

Answer 1

We are allowed to do only 1 question at a time.

a)

The Central Limit Theorem (CLT) is a statistical theory states that given a sufficiently large sample size from a population with a finite level of variance, the mean of all samples from the same population will be approximately equal to the mean of the population.

As the sample size increases, the sampling distribution of the mean, X-bar, can be approximated by a normal distribution with mean µ and standard deviation σ/√n where:

µ is the population mean
σ is the population standard deviation
n is the sample size

In other words, if we repeatedly take independent random samples of size n from any population, then when n is large, the distribution of the sample means will approach a normal distribution.