Sampling Mean and Sample Size Relationship

E

The central limit theorem states that as the sample size gets larger, the sampling distribution of the sample means becomes approximately normal.

As the size of the sample increases, the shape of the sampling mean tends to approach a normal distribution.

The sampling mean refers to the average value calculated from a sample of data points taken from a larger population. When a sample is drawn from a population, the sampling mean can vary depending on which data points are included in the sample. However, as the sample size increases, the sampling mean tends to become more stable and better represents the population mean.

According to the Central Limit Theorem (CLT), when independent random variables are added, their sum tends toward a normal distribution, regardless of the shape of the original distribution, as long as the sample size is sufficiently large. This means that the sampling mean, which is the sum of the sample values divided by the sample size, will also tend toward a normal distribution as the sample size increases.

The normal distribution, also known as the Gaussian distribution or bell curve, is a symmetric probability distribution that is characterized by its mean and standard deviation. It is widely observed in nature and commonly used in statistical analyses. The shape of a normal distribution is bell-shaped, with the mean located at the center and the data symmetrically distributed around it.

Therefore, as the sample size increases, the shape of the sampling mean becomes more bell-shaped and symmetric, approaching a normal distribution. This phenomenon holds true regardless of the shape of the original population distribution from which the sample is drawn. Thus, the correct answer to the question is option E: Approaches a normal distribution.