Question

question 5
2 pts
for a population with a mean of $mu = 40$ and a standard deviation of $sigma = 8$, find the z - score corresponding to each of the following samples:
$m = 34$ for a sample of $n = 4$ scores
$sigma_m=$ and $z=$
$m = 34$ for a sample of $n = 16$ scores
$sigma_m=$ and $z=$

Explanation:

Step1: Calculate standard error for n = 4

The formula for the standard error of the mean $\sigma_M=\frac{\sigma}{\sqrt{n}}$. Given $\sigma = 8$ and $n = 4$, we have $\sigma_M=\frac{8}{\sqrt{4}}$.
$\sigma_M=\frac{8}{2}=4$

Step2: Calculate z - score for n = 4

The formula for the z - score of a sample mean is $z=\frac{M-\mu}{\sigma_M}$. Given $\mu = 40$, $M = 34$ and $\sigma_M = 4$, we have $z=\frac{34 - 40}{4}$.
$z=\frac{-6}{4}=-1.5$

Step3: Calculate standard error for n = 16

Using the formula $\sigma_M=\frac{\sigma}{\sqrt{n}}$, with $\sigma = 8$ and $n = 16$, we get $\sigma_M=\frac{8}{\sqrt{16}}$.
$\sigma_M=\frac{8}{4}=2$

Step4: Calculate z - score for n = 16

Using the formula $z=\frac{M-\mu}{\sigma_M}$, with $\mu = 40$, $M = 34$ and $\sigma_M = 2$, we have $z=\frac{34 - 40}{2}$.
$z=\frac{-6}{2}=-3$

Answer:

For $n = 4$, $\sigma_M = 4$ and $z=-1.5$
For $n = 16$, $\sigma_M = 2$ and $z=-3$