Question

What is the mean of the sampling distribution of the sample mean $x$ if a sample of $64$ students is selected at random from the entire freshman class? The dean of admissions in a large university has determined that the scores of the freshman class in a mathematics test are normally distributed with a mean of $82$ and a standard deviation of $8$.

Answer 1

Step-1: Given data:

Given that,

The total number of students or sample size $n=64$.

The population mean, $\mu =82$.

The standard deviation of ${\sigma }_2=8$.

Let $X$ be a normal random variable.

Step-2: Apply the normal random variable formula:

Use the formula $X~N(\mu ,{\sigma }_2)$

Substitute the known values in the above formula:

$\Rightarrow X~N(82,8)$

Since, the population distribution is normal and also sample size is greater than $30$, the sampling distribution of the sample mean is approximately normal.

Step-3: Find the mean of the sampling distribution of the sample mean:

To find the mean and standard deviation of the sampling distribution use the formula $\left({\overline{\mu }}_x\right)\times \left({\overline{\sigma }}_x\right)$. Here ${\overline{\mu }}_x$ is the mean and ${\overline{\sigma }}_x$ is the standard deviation.

Calculate the mean of the sampling distribution:

$\Rightarrow {\overline{\mu }}_x=\mu =82$

Calculate the standard deviation of the sampling distribution:

$\begin{array}{rcl}{\overline{\sigma }}_x& =& \frac{\sigma }{\sqrt{n}}\\ & =& \frac{8}{\sqrt{64}}\\ & =& \frac{8}{8}\\ & =& 1\end{array}$

Substitute the mean and standard deviation values in the formula $\left({\overline{\mu }}_x\right)\times \left({\overline{\sigma }}_x\right)$:

$\Rightarrow 82\times 1=82$

Hence, the mean of the sampling distribution of the sample mean is$\mathbf{82}\mathbf{.}$