Question

find the sample variance and standard deviation.
7, 46, 14, 51, 38, 22, 31, 27, 35, 26
choose the correct answer below. fill in the answer box to complete your choice.
(round to two decimal places as needed.)
a. $s^{2}=184.46$
b. $sigma^{2}=$
choose the correct answer below. fill in the answer box to complete your choice.
(round to one decimal place as needed.)
a. $sigma=$
b. $s=$

Explanation:

Step1: Calculate sample mean

First, sum all data points and divide by sample size $n=10$.
$\bar{x} = \frac{7+46+14+51+38+22+31+27+35+26}{10} = \frac{297}{10} = 29.7$

Step2: Compute squared deviations

Find each $(x_i - \bar{x})^2$:
$(7-29.7)^2=515.29$, $(46-29.7)^2=265.69$, $(14-29.7)^2=246.49$, $(51-29.7)^2=453.69$, $(38-29.7)^2=68.89$, $(22-29.7)^2=59.29$, $(31-29.7)^2=1.69$, $(27-29.7)^2=7.29$, $(35-29.7)^2=28.09$, $(26-29.7)^2=13.69$

Step3: Sum squared deviations

Add all the squared values:
$515.29+265.69+246.49+453.69+68.89+59.29+1.69+7.29+28.09+13.69 = 1650.1$

Step4: Calculate sample variance

Use sample variance formula $s^2=\frac{\sum(x_i-\bar{x})^2}{n-1}$:
$s^2 = \frac{1650.1}{10-1} \approx 183.34$
(Note: The pre-selected 184.46 is incorrect; correct sample variance is calculated here)

Step5: Calculate sample standard deviation

Take square root of sample variance:
$s = \sqrt{183.34} \approx 13.5$

Answer:

For variance:
A. $s^2 = 183.34$
For standard deviation:
B. $s = 13.5$