A law school with a total enrollment of 5000 students is known to have 40% women in its student body.  Suppose we take a random sample of 100 students from the law school.   What value should we expect for our sample proportion of women?  What is the standard error?  According to the Central Limit Theorem should I expect the sampling distribution of the sample proportion to be approximately Normal?

Question

A law school with a total enrollment of 5000 students is known to have 40% women in its student body.  Suppose we take a random sample of 100 students from the law school.   What value should we expect for our sample proportion of women?  What is the standard error?  According to the Central Limit Theorem should I expect the sampling distribution of the sample proportion to be approximately Normal?

Step 1

Expectation:

Denote π as the true population proportion of women in the law school. Hence, π = 0.4.

Denote p as the sample proportion of women in the law school.

The sample proportion is an unbiased estimator of the population proportion. So, whatever be the distribution of p, the expected value of the sample proportion is: E (p) = π = 0.4.

Standard error:

The population size here, is N = 5,000. The sample size is n = 100.

While calculating the standard error of the sample proportion in case of a sampling without replacement, the “finite population correction factor” must be included, if the sample size is more than 5% of the population.

Now, if we collect a sample of students out of all the students in the law school, we do not want our sample to consider the same student more than once; the sample should consist of unique individuals. In other words, we want the sampling to be without replacement.

Now, 5% of 5,000 = 250. Evidently, our sample size is less than 5% of the population size. In other words, the population can be considered as effectively infinite as compared to the sample size. Therefore, no finite population correction factor is needed for calculating the standard error.

The standard error of the sample proporti...

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