1 Answer

Gopikrishna S · Answer 1 · 2024-07-25T22:45:57+0000

A.
$s^2 = 243.17$ (rounded to two decimal places)

B.
$\sigma^2$ is not applicable in this case since it's a sample, and the correct notation for the sample variance is
$s^2$ .

How did we get the value?

To find the sample variance
$($s^2$)$ and standard deviation
$($s$)$ , you can follow these steps:

1. Find the mean
$($x-bar}$)$ of the data set.

2. Subtract the mean from each data point and square the result.

3. Sum up all the squared differences.

4. Divide the sum by
$n-1$ for the sample variance
$($s^2$)$ , where
$n$ is the number of data points.

5. Take the square root of the sample variance to find the sample standard deviation
$($s$)$ .

Calculate it for the given data set:

Data set: 4, 57, 12, 50, 34, 26, 29, 30, 35, 28

1. Mean
$($x-bar}$)$ :

$x-bar} = (4 + 57 + 12 + 50 + 34 + 26 + 29 + 30 + 35 + 28)/(10) = (305)/(10) = 30.5$

2. Squared differences from the mean:

$(4-30.5)^2, (57-30.5)^2, (12-30.5)^2, (50-30.5)^2, (34-30.5)^2, (26-30.5)^2, (29-30.5)^2, (30-30.5)^2, (35-30.5)^2, (28-30.5)^2$

3. Sum of squared differences:

$676.25 + 676.25 + 342.25 + 361.25 + 10.25 + 20.25 + 2.25 + 0.25 + 20.25 + 6.25 = 2188.5$

4. Sample variance
$($s^2$)$ :

$s^2 = (2188.5)/(10-1) = (2188.5)/(9) \approx 243.166$

5. Sample standard deviation
$($s$)$ :

$s = √(243.166) \approx 15.59$

Therefore, the answers are:

A.
$s^2 = 243.17$ (rounded to two decimal places)

B.
$\sigma^2$ is not applicable in this case since it's a sample, and the correct notation for the sample variance is
$s^2$ .

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