Question

question 4
3 pts
for a sample of n = 16 scores, what is the value of a population standard deviation (σ) necessary to produce each of the following standard error values?
if the standard deviation is 32, then σ_m=
if the standard deviation is 16, then σ_m=
if the standard deviation is 4, then σ_m=

Explanation:

Step1: Recall standard - error formula

The formula for the standard error of the mean $\sigma_M=\frac{\sigma}{\sqrt{n}}$, where $\sigma$ is the population standard deviation and $n$ is the sample size. Given $n = 16$, so $\sqrt{n}=4$.

Step2: Calculate $\sigma_M$ when $\sigma = 32$

Substitute $\sigma = 32$ and $\sqrt{n}=4$ into the formula $\sigma_M=\frac{\sigma}{\sqrt{n}}$. Then $\sigma_M=\frac{32}{4}=8$.

Step3: Calculate $\sigma_M$ when $\sigma = 16$

Substitute $\sigma = 16$ and $\sqrt{n}=4$ into the formula $\sigma_M=\frac{\sigma}{\sqrt{n}}$. Then $\sigma_M=\frac{16}{4}=4$.

Step4: Calculate $\sigma_M$ when $\sigma = 4$

Substitute $\sigma = 4$ and $\sqrt{n}=4$ into the formula $\sigma_M=\frac{\sigma}{\sqrt{n}}$. Then $\sigma_M=\frac{4}{4}=1$.

Answer:

If the standard deviation is 32, then $\sigma_M = 8$.
If the standard deviation is 16, then $\sigma_M = 4$.
If the standard deviation is 4, then $\sigma_M = 1$.