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2 The random variable
X has mean 372 and standard deviation 54 .
(i) Describe fully the distribution of the mean of a random sample of 36 values of
X.

$2$ The random variable $X$ has mean $372$ and standard deviation $54$ . $\newline$ (i) Describe fully the distribution of the mean of a random sample of $36$ values of $X$ .

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Find values of normal variables

Full solution

Q.

2

The random variable

X

has mean

372

and standard deviation

54

.

\newline

(i) Describe fully the distribution of the mean of a random sample of

36

values of

X

.

Identify Mean of X: Step $1$ : Identify the mean of the random variable $X$ . The mean ( $\mu$ ) of $X$ is given as $372$ .
Identify Standard Deviation: Step $2$ : Identify the standard deviation of the random variable $X$ . The standard deviation ( $\sigma$ ) of $X$ is given as $54$ .
Calculate Sample Standard Deviation: Step $3$ : Calculate the standard deviation of the mean of the sample. $\newline$ Since the sample size $n$ is $36$ , the standard deviation of the sample mean $\sigma_{\text{mean}}$ is calculated using the formula: $\newline$ $\sigma_{\text{mean}} = \frac{\sigma}{\sqrt{n}}$ $\newline$ $\sigma_{\text{mean}} = \frac{54}{\sqrt{36}}$ $\newline$ $\sigma_{\text{mean}} = \frac{54}{6}$ $\newline$ $\sigma_{\text{mean}} = 9$
Describe Sample Mean Distribution: Step $4$ : Describe the distribution of the sample mean. The distribution of the sample mean is normally distributed (by the Central Limit Theorem) with mean $\mu = 372$ and standard deviation $\sigma_{\text{mean}} = 9$ .

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