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 = 70, find the probability of a sample mean being less than 25.1 if μ = 25 and σ = 1.23. click the icon to view page 1 of the standard normal table. click the icon to view page 2 of the standard normal table. for a sample of n = 70, the probability of a sample mean being less than 25.1 if μ = 25 and σ = 1.23 is (round to four decimal places as needed.)

Explanation:

Step1: Calculate the standard error

The formula for the standard error of the mean is $\sigma_{\bar{x}}=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 1.23$ and $n = 70$, we have $\sigma_{\bar{x}}=\frac{1.23}{\sqrt{70}}\approx\frac{1.23}{8.3666}\approx0.147$.

Step2: Calculate the z - score

The z - score formula for the sample mean is $z=\frac{\bar{x}-\mu}{\sigma_{\bar{x}}}$. Here, $\bar{x}=25.1$, $\mu = 25$ and $\sigma_{\bar{x}}\approx0.147$. So $z=\frac{25.1 - 25}{0.147}=\frac{0.1}{0.147}\approx0.68$.

Step3: Find the probability

We want to find $P(\bar{X}<25.1)$, which is equivalent to $P(Z < 0.68)$ using the standard normal distribution. Looking up the value in the standard - normal table, $P(Z < 0.68)=0.7517$.

Answer:

$0.7517$