Question

In a large population of adults, the mean IQ is 112 with standard deviation 20. The IQ values are normally distributed. Suppose 20 adults are randomly selected for a market research campaign.

A) Verify that the sample means computed from samples of size 20 meets the conditions that its sampling distribution be approximately normal?

Answer 1

Since the population IQ values are normally distributed, the sample means computed from samples of size 20 will have a sampling distribution that is approximately normal, according to the Central Limit Theorem.

Step-by-step explanation:

To verify that the sampling distribution of the sample mean for a sample size of 20 is approximately normal, we can use the Central Limit Theorem (CLT). The CLT states that the sampling distribution of the sample mean will be normally distributed if the sample size is sufficiently large, typically n ≥ 30, or if the population distribution is normal for smaller samples.

In this scenario, the population mean IQ is given as 112 with a standard deviation of 20, and the population is described as normally distributed. Since we are dealing with an underlying normal distribution, the sampling distribution of the sample mean for a sample size of 20 will also be approximately normal. This meets one of the conditions set by the CLT for smaller sample sizes.

Therefore, for market research purposes, we can assume that the sampling distribution of the sample mean will be approximately normal for samples of size 20 from this population of adults.