As the size of the sample increases, what happens to the shape of the sampling distribution of sample means?
A. It cannot be predicted in advance. B. It approaches a normal distribution. C. It is positively skewed. D. It is negatively skewed.
The correct answer and explanation is:
The correct answer is B. It approaches a normal distribution.
When the sample size increases, the shape of the sampling distribution of sample means becomes more closely aligned with a normal distribution, regardless of the shape of the population distribution. This concept is a result of the Central Limit Theorem (CLT), which states that for any population with a finite mean and variance, the distribution of sample means will approach a normal distribution as the sample size increases, typically when the sample size exceeds 30.
Here’s a breakdown of the process:
- Sample Means Distribution: Initially, if the population is not normally distributed, the distribution of sample means may reflect the shape of the population. For example, if the population is skewed, the distribution of sample means will also be skewed when the sample size is small.
- Increasing Sample Size: As the sample size increases, the sampling distribution of the sample means becomes more symmetric and bell-shaped, eventually resembling a normal distribution. This occurs because the larger the sample, the less likely random fluctuations in the data will create skewed results, and the law of large numbers ensures that the sample mean will tend to be close to the population mean.
- Role of Central Limit Theorem: The CLT is crucial because it allows researchers to make inferences about the population from sample means, even when the underlying population distribution is not normal. It provides a foundation for hypothesis testing and confidence intervals, which rely on the normality of the sampling distribution.
- Key Point: This convergence to normality happens as the sample size grows, making the sampling distribution of sample means approximately normal for sufficiently large samples, even if the population itself is not normally distributed.
Thus, B is the correct answer because the sampling distribution of sample means tends to become more normal as the sample size increases.