Question

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual For a sample of a 70 find the probably of a sample mean being greater than 213 2 12 and 5.9 For a sample 70, the probability of a sample mean being greater than 213 Pound to four decimal places as needed) 212 and 595 Would the given Sample mean be considered unusual? The sample mean be considered unusual because it Mwiring of autuvert, amely within of the mean of the sample means

Answer 1

Let X be a random variable with  mean $\mu=212$ and standard deviation $\sigma=5.9$

Let $\bar{X}$ be the sample mean of a randomly selected sample of size n=70 from the above population.

Since the sample size is greater than 30, using the central limit theorem, we can say that the distribution of  $\bar{X}$ is normal distribution with mean $\mu_{\bar{X}}=\mu=212$ and standard deviation (also called standard error of mean) $\sigma_{\bar{X}}=\frac{\sigma}{\sqrt{n}}=\frac{5.9}{\sqrt{70}}=0.7052$

The probability of a sample mean being greater than 213 is

$\begin{align*} P(\bar{X}>213 )&=P\left(\frac{\bar{X}-\mu_{\bar{X}}}{\sigma_{\bar{X}}}>\frac{213-\mu_{\bar{X}}}{\sigma_{\bar{X}}} \right )\quad\text{get the z score of 213}\\ &=P\left(Z>\frac{213-212}{0.7052} \right )\\ &=P(Z>1.42)\\ &=1-P(Z<1.42)\\ &=1-0.9222\quad\text{using the standard normal tables for z=1.42}\\ &=0.0778 \end{align*}$

ans: For a sample of n=70, the probability of a sample mean being greater than 213 is 0.0778

We consider an event unusual if the corresponding value is within +-2 standard deviations of mean. Here, the z score of the sample mean 213 is 1.42 and hence 213 lies within 2 standard deviations from the mean. Hence we do not consider this an unusual event.

ans: The sample mean not to be considered unusual because it lies within the range of a usual event, namely within 2 standard deviations of the mean of the sample means.