Question

for the following situation, find the mean and standard deviation of the population. list all samples (with replacement) of the given size from that population and find the mean of each. find the mean and standard deviation of the sampling distribution and compare them with the mean and standard deviation of the population. the scores of three students in a study group on a test are 91, 90, 92. use a sample size of 3. the mean of the population is 91.00 (round to two decimal places as needed.) the standard deviation of the population is (round to two decimal places as needed.)

Explanation:

Step1: Recall standard - deviation formula

The formula for the population standard deviation $\sigma=\sqrt{\frac{\sum_{i = 1}^{N}(x_{i}-\mu)^{2}}{N}}$, where $x_{i}$ are the data points, $\mu$ is the population mean, and $N$ is the number of data points. Here, $N = 3$, $\mu=91$, $x_1 = 91$, $x_2=90$, $x_3 = 92$.

Step2: Calculate $(x_{i}-\mu)^{2}$ for each data - point

For $x_1 = 91$: $(91 - 91)^{2}=0$; for $x_2 = 90$: $(90 - 91)^{2}=1$; for $x_3 = 92$: $(92 - 91)^{2}=1$.

Step3: Calculate the sum of $(x_{i}-\mu)^{2}$

$\sum_{i = 1}^{3}(x_{i}-\mu)^{2}=0 + 1+1=2$.

Step4: Calculate the population standard deviation

$\sigma=\sqrt{\frac{2}{3}}\approx0.82$.

Answer:

$0.82$