Question

4. Explain the difference between the sample mean and the mean of the sampling distribution.

Answer 1

See explanation below.

Explanation:

The sample mean is just a number calculated from the following formula:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

Assuming n observation
`x_1, x_2,.....,x_n`

By the other hand the sampling distribution of the mean is the distribution for possible sample means that we take from the population, and when the sample size is large enough (n>30) we can use the central limit theorem to find the distribution for the sample mean.

The central limit theorem states that "if we have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large".

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

From the central limit theorem we know that the distribution for the sample mean
`\bar X` is given by if n>30:

`\bar X \sim N(\mu, (\sigma)/(√(n)))`