According To The Central Limit Theorem, A Population Is

1according To The Central Limit Theorem A Population Which Is Skewed

According to the central limit theorem, a population which is skewed to begin with will still be skewed when it is re-formed as a distribution of sample means. (Points : 1)

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The Central Limit Theorem (CLT) is a fundamental principle in statistics that explains the behavior of the sampling distribution of the sample mean. It states that, regardless of the shape of the population distribution, the distribution of the sample means will tend to be approximately normal if the sample size is sufficiently large. This theorem is crucial because it allows statisticians to make inferences about populations using normal distribution techniques, even when the underlying population distribution is skewed, bimodal, or otherwise non-normal.

However, an important nuance of the CLT pertains to populations that are initially skewed. When a population distribution is skewed—meaning it is asymmetrical with a longer tail on one side—the distribution of the sample means derived from that population does not remain skewed regardless of sample size. Instead, the degree of skewness in the sampling distribution diminishes as the sample size increases. Specifically, for small sample sizes, the distribution of the sample mean can still exhibit significant skewness. This residual skewness reflects the skewness of the population, albeit somewhat less pronounced.

As the sample size grows larger, the effect of skewness in the population diminishes in the sampling distribution. According to the CLT, when the sample size is sufficiently large (often n ≥ 30 is used as a rule of thumb), the distribution of the sample means approaches a normal distribution, which is symmetric. This convergence occurs regardless of skewness in the original population. Consequently, for large samples, the distribution of the sample mean tends to be approximately normal, even if the underlying population distribution is skewed.

This phenomenon has practical implications in statistical analysis and hypothesis testing. It means that practitioners should consider the size of the samples they are using when applying normal-theory methods to skewed data. For small samples from a skewed population, the sampling distribution may not be approximately normal, and adjustments or non-parametric methods might be more appropriate. Conversely, with large samples, the CLT assures that the distribution of the sample mean will be nearly normal, facilitating the use of standard inferential techniques.

To summarize, the statement that a population which is skewed will still be skewed when re-formed as a distribution of sample means is generally not accurate for large sample sizes. While small samples retain more of the population’s skewness, the CLT guarantees that the distribution of the sample mean becomes increasingly normal as the sample size increases. Therefore, the skewness diminishes, and the distribution approaches a symmetric normal curve, making statistical inference more straightforward.

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