Respuesta :
Adding 10 to every observation in a sample, the sample mean would be 10 more the precious one and the standard deviation will remain the same.
What is mean of sample?
"It is an average of a set of data."
What is formula of mean of sample?
"For a sample of observation [tex]x_1,x_3,x_2,...,x_n[/tex] the mean of the sample is,
[tex]\Rightarrow \bar{X}=\frac{x_1+x_3+x_2+...+x_n}{n}[/tex]"
What is standard deviation?
"It is a measure of how dispersed the data from the mean."
Formula for standard deviation?
[tex]\Rightarrow \sigma=\sqrt{\frac{\Sigma(x_i-\bar{X})^2}{N}}[/tex]
where, [tex]\bar{X}[/tex] is the mean of sample and N is the size of sample
For given question,
Let [tex]x_1,x_3,x_2,...,x_n[/tex] be a sample observation.
We need to check if we add 10 to all of the observations in a sample, then does this change the sample mean. Also we have to check how does it change the sample standard deviation.
Let [tex]\bar{X}[/tex] be the mean of [tex]x_1,x_3,x_2,...,x_n[/tex]
[tex]\Rightarrow \bar{X}=\frac{x_1+x_3+x_2+...+x_n}{n}[/tex]
We add 10 to all of the observations in a sample.
[tex]\Rightarrow x_1+10,x_3+10,x_2+10,...,x_n+10[/tex]
The mean of above sample would be,
[tex]=\frac{(x_1+10)+(x_3+10)+(x_2+10)...+(x_n+10)}{n}\\\\=\frac{10n + (x_1+x_3+x_2+...+x_n)}{n}\\\\ =\frac{10n}{n} +\frac{x_1+x_3+x_2+...+x_n}{n}\\\\ =10 + \bar{X}[/tex]
This means, if we add 10 to every observation in a sample then the sample mean we 10 more the precious one.
In case of standard deviation,
adding 10 to every observation will not change the [tex](x_i-\bar{X})[/tex] factor.
This means, the standard deviation will remain the same.
Therefore, adding 10 to every observation in a sample, the sample mean would be 10 more the precious one and the standard deviation will remain the same.
Learn more about mean and standard deviation here:
https://brainly.com/question/16030790
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