Question

According to the FDA, 76% of all people are omnivores. We will use sampling distributions here to determine what kinds of samples we can expect based on this data using sampling distributions.

a) Using a sample of size 20, find the point estimate you got creating a sampling distribution with 3000 samples. Also find the standard error.

b) Using a sample of size 20, find the point estimae and standard error for 6000 samples instead. Compare this to your results from a.

c) Using a sample of size 50, find the point estimate and standard error still using 3000 samples. Compare this with your results from a

d) Using a sample of size 100, find the point estimae and standard error for 3000 samples instead. Compare this to your results from a and c.

e) What conclusions can we make about sampling distributions concerning the point estimate and standard error based on the sample size and the number of samples.

Answer 1

The point estimate remains stable regardless of the number of samples taken, and it typically mirrors the known population proportion. The standard error decreases as the sample size increases, implying that the precision of the estimate improves with larger samples, which aligns with the Central Limit Theorem.

Step-by-step explanation:

The student's question revolves around the concept of sampling distributions and how varying the sample size and number of samples can affect the point estimate and standard error.

When calculating the point estimate for a given number of samples (3000 or 6000), it is essentially the sample mean of all samples, and it should be very close to the population proportion given by the FDA, which is 76% in this case. Increasing the number of samples doesn't affect the point estimate significantly, as the sample mean is an unbiased estimate of the population mean.

The standard error reflects how much sample means will differ from the population mean. It decreases when the sample size increases but is not affected by the number of samples. For instance, when using a sample size of 20, the standard error will be larger than when using a sample size of 100, presuming the population standard deviation is constant. The formula for standard error (SE) of a proportion is SE = sqrt[(p(1-p))/n], where p is the population proportion and n is the sample size.

As the sample size increases, point estimates will remain relatively stable, but the standard error will decrease, indicating that the estimates will be more precise. This corresponds to the Central Limit Theorem, which asserts that as sample size grows, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the population's distribution.