Question

Calculate the range, population variance, and population standard deviation for the following data set. If necessary, round to one more decimal place than the largest number of decimal places given in the data. 16,12,17,8,15,15,10,11,19,18

Answer 1

`Range = 11`

`\sigma^2 = 12.1`

`\sigma = 3.5`

Explanation:

Given

`Data: 16,12,17,8,15,15,10,11,19,18`

Solving (a): Range

This is calculated as:

`Range = Highest - Least`

Where:

`Highest = 19`

`Least = 8`

So:

`Range = 19 - 8`

`Range = 11`

Solving (b): The population variance

First, calculate the population mean using:

`\mu = (\sum x)/(n)`

So:

`\mu = (16+12+17+8+15+15+10+11+19+18)/(10)`

`\mu = (141)/(10)`

`\mu = 14.1`

So, the population variance is:

`\sigma^2 = (\sum(x - \mu)^2)/(n)`

`\sigma^2 = ((16 - 14.1)^2 + (12 - 14.1)^2 +............... + (19- 14.1)^2 + (18- 14.1)^2)/(10)`

`\sigma^2 = (120.9)/(10)`

`\sigma^2 = 12.09`

`\sigma^2 = 12.1` --- approximated

Solving (c): The population standard deviation.

This is calculated as:

`\sigma = \sqrt{\sigma^2`

`\sigma = \sqrt{12.09`

`\sigma = 3.5`