The Central Limit Theorem (CLT) implies that:
A. the population will be approximately normal if n = 30.
B. repeated samples must be taken to obtain normality.
C. the distribution of the mean is approximately normal for large n.
D. the mean follows the same distribution as the population.
The correct answer and explanation is :
The correct answer is:
C. the distribution of the mean is approximately normal for large n.
Explanation:
The Central Limit Theorem (CLT) is one of the fundamental theorems in statistics, particularly important when working with sampling distributions. It states that, regardless of the original population distribution, the distribution of the sample means will approximate a normal distribution as the sample size (n) becomes sufficiently large.
This principle is especially useful when dealing with populations that do not follow a normal distribution. The CLT allows us to apply statistical methods that assume normality (such as hypothesis testing or confidence intervals), even when the population data is skewed or non-normal, provided the sample size is large enough.
Why answer C is correct:
Option C states that “the distribution of the mean is approximately normal for large n,” which is the essence of the CLT. The CLT implies that as the sample size increases (usually n ≥ 30 is considered large enough), the distribution of the sample means will tend to a normal distribution, no matter what the shape of the population distribution is.
Why other options are incorrect:
- A. “The population will be approximately normal if n = 30.”
This is incorrect because the CLT applies to the distribution of sample means, not the population itself. The population distribution does not need to be normal for the CLT to hold. It’s the sample means that become normal as sample size increases. - B. “Repeated samples must be taken to obtain normality.”
This is misleading. While repeated sampling is part of the process of studying sampling distributions, the CLT states that the sample mean distribution becomes approximately normal as the sample size increases, not due to repeated sampling. - D. “The mean follows the same distribution as the population.”
This is incorrect because the distribution of the sample mean is different from the population distribution. The sample mean is typically less spread out (has lower variance) than the population, and as sample size increases, the sample mean distribution becomes more concentrated around the population mean.
In summary, the Central Limit Theorem emphasizes that as the sample size increases, the distribution of sample means tends toward normality, which is key in many statistical techniques.