Question

what does the central limit theorem say about the shape of the distribution of the sample means?
the distribution will be approximately normal when the sample size is small.
the distribution will be approximately normal when the standard deviation of the
sample means is the same as the population standard deviation.
the distribution will be approximately normal when the sample size is large.
the distribution will be normal when all sample means within 2 standard
deviations of the actual mean are considered.
the distribution will be approximately normal regardless of sample size.

Explanation:

Step1: Recall Central Limit Theorem

The Central Limit Theorem states that if you have a population with mean \(\mu\) and standard deviation \(\sigma\) and take sufficiently large random samples from the population with replacement, the distribution of the sample means will be approximately normally distributed.

Step2: Analyze Each Option

• Option A: The distribution is approximately normal for large sample sizes, not regardless of sample size.
• Option B: The standard deviation of sample means (\(\frac{\sigma}{\sqrt{n}}\)) is different from population standard deviation \(\sigma\) (unless \(n = 1\)).
• Option C: This is in line with the Central Limit Theorem. As \(n\) (sample size) gets large (usually \(n\geq30\)), the sampling distribution of the sample mean \(\bar{X}\) is approximately normal.
• Option D: The actual mean of sample means is \(\mu\) (same as population mean), but this is about the mean, not the shape related to the Central Limit Theorem.
• Option E: A normal distribution within 2 - standard deviations is a property of normal distributions in general, not specifically what the Central Limit Theorem emphasizes about the shape of the sampling distribution of the sample mean.

Answer:

C. The distribution will be approximately normal when the sample size is large.