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 n = 75, find the probability of a sample mean being greater than 212 if μ = 211 and σ = 6.1. for a sample of n = 75, the probability of a sample mean being greater than 212 if μ = 211 and σ = 6.1 is 0.0778 (round to four decimal places as needed.) would the given sample mean be considered unusual? the sample mean be considered unusual because it within the range of a usual event, namely within of the mean of the sample means.

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$, where $\sigma = 6.1$ and $n = 75$. So, $\sigma_{\bar{x}}=\frac{6.1}{\sqrt{75}}\approx\frac{6.1}{8.66}\approx0.704$.

Step2: Calculate the z - score

The z - score formula for the sample mean is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$, with $\bar{x} = 212$, $\mu=211$ and $\sigma_{\bar{x}}\approx0.704$. So, $z=\frac{212 - 211}{0.704}\approx1.42$.

Step3: Find the probability

We want $P(\bar{X}>212)$, which is equivalent to $P(Z > 1.42)$ in the standard normal distribution. Since $P(Z>z)=1 - P(Z\leq z)$, and from the standard - normal table $P(Z\leq1.42)=0.9222$, then $P(Z > 1.42)=1 - 0.9222 = 0.0778$.

Step4: Determine if the sample mean is unusual

A usual event has a probability between 0.05 and 0.95 for values within 2 standard deviations of the mean. Since $P(\bar{X}>212)=0.0778>0.05$, the sample mean is not unusual. It is within 2 standard deviations of the mean of the sample means.

Answer:

The sample mean would not be considered unusual because it is within 2 standard deviations of the mean of the sample means.