Question

a simple random sample of size n=40 is obtained from a population with a mean of 20 and a standard deviation of 5. is the sampling distribution normally distributed? why?

yes, the sampling distribution is normally distributed because the sample size is greater than 30.
yes, the sampling distribution is normally distributed because the population is normally distributed.
no, the sampling distribution is not normally distributed because the population is not normally distributed.
no, the sampling distribution is not normally distributed because the population mean is less than 30.

Explanation:

Identify the sample size

Using the Sampling Distribution knowledge point

$$ n = 40 $$

Apply the Central Limit Theorem

Using the Central Limit Theorem knowledge point

$$ n = 40 \ge 30 $$

The sampling distribution of the sample mean is approximately normal regardless of the population distribution shape.

Evaluate the given options

Using the Central Limit Theorem and Sampling Distribution knowledge points

• Option 1: Correct. The sample size \(n = 40\) is greater than 30.
• Option 2: Incorrect. The population distribution shape is not specified.
• Option 3: Incorrect. The Central Limit Theorem guarantees normality for large samples.
• Option 4: Incorrect. Normality depends on sample size, not the population mean.

Answer:

• (A) Yes, the sampling distribution is normally distributed because the sample size is greater than 30. (Correct answer)
• (B) Yes, the sampling distribution is normally distributed because the population is normally distributed.
• (C) No, the sampling distribution is not normally distributed because the population is not normally distributed.
• (D) No, the sampling distribution is not normally distributed because the population mean is less than 30.