Question

A sample of 96 observations is taken from an infinite population. The sampling distribution of xˉ is approximately normal because xˉ is always approximately normally distributed. normal because of the central limit theorem. normal because the sample size is small in comparison to the population size. None of these alternatives is correct.

Answer 1

The sampling distribution of xˋ is normal because of the central limit theorem, which applies due to the sufficiently large sample size of 96.

Step-by-step explanation:

The answer to the student's question is that the sampling distribution of â‒ᦇ is normal because of the central limit theorem. The central limit theorem (CLT) states that the distribution of sample means will approach normality as the sample size increases, regardless of the population's distribution. Given that the sample size in the question is 96, which is greater than 30, it is sufficiently large for the CLT to apply.

According to the CLT, the mean of the sampling distribution will be equal to the population mean, and the standard deviation of the sampling distribution will be the population standard deviation divided by the square root of the sample size. Therefore, the fact that the sample is a small portion of the population does not make the distribution normal - it is the relatively large sample size that validates the use of the CLT to assume normality of the sampling distribution of the mean.