Question

For the following set of data, find the number of data within 2 population standard deviations of the mean.28, 65, 114, 74, 68, 75, 70, 69, 64

Answer 1

To determine the data that is within 2 population standard deviations of the mean, let's calculate the mean first.

To determine the mean, let's add all the data and divide the result by the total number of data.

`28+65+114+74+68+75+70+69+64=627`
`627/9=69.66667`

The mean is 69.66667.

Let's now calculate the standard deviation. Here are the steps:

1. Subtract the mean from each data, then square the result.

`\begin{gathered} 28-69.66667=(-41.66667)^2=1,736.1114 \\ 65-69.66667=(-4.66667)^2=21.7778 \end{gathered}`
`\begin{gathered} 114-69.66667=(44.33333)^2=1,965.4441 \\ 74-69.66667=(4.33333)^2=18.7777 \end{gathered}`
`\begin{gathered} 68-69.66667=(-1.66667)^2=2.7778 \\ 75-69.66667=(5.33333)^2=28.4444 \end{gathered}`
`\begin{gathered} 70-69.66667=(0.33333)^2=0.1111 \\ 69-69.66667=(-0.66667)^2=0.4444 \\ 64-69.66667=(-5.66667)^2=32.1111 \end{gathered}`

2. Add the results in step 1.

`1,736.1114+21.7778+1,965.4441+18.7777+2.7778=3,744.8888`
`28.4444+0.1111+0.4444+32.1111=61.111`
`3,744.8888+61.111=3,805.9998`

The sum is 3, 805.9998.

3. Divide the sum by the total number of data.

`3,805.9998/9=422.8889`

4. Square root the result in step 3.

`√(422.8889)\approx20.56`

The standard deviation is approximately 20.56.

So, the data that are within 2 population standard deviations of the mean are between:

`\begin{gathered} 69.67-(2)(20.56)=28.55\approx29 \\ 69.67+(2)(20.56)=110.79\approx111 \end{gathered}`

The data that are within 2 population standard deviations of the mean are between 29 and 111. Based on the given data, the data that are between 29 and 111 are the following: 64, 65, 68, 69, 70, 74, and 75. There are 7 data that are within 2 population standard deviations of the mean.