Variance & Standard Deviation | Introduction to Statistics and Data Analysis | Probability & Statistics for Engineers & Scientists | Walpole | Problem 1.9

Solution:

For the samples with NO AGING:

We know, based on our answer in Exercise 1.3, that the sample mean for samples with no aging is $\displaystyle \overline{x}_{no\:aging}=222.10$.

To compute for the sample variance, we shall use the formula

$\displaystyle s^2=\sum _{i=1}^n\frac{\left(x_i-\overline{x}\right)^2}{n-1}$

The formula states that we need to get the sum of $\displaystyle \left(x_i-\overline{x}\right)^2$, so we can use a table to solve $\displaystyle \left(x_i-\overline{x}\right)^2$ for every sample.

Note: You can refer to the solution of Exercise 1.7 on how to use a table to solve for the variance.

The variance is

$\displaystyle \left(s^2\right)_{no\:aging}=\sum _{i=1}^{10}\:\frac{\left(x_i-\overline{x}\right)^2}{n-1}$

$\displaystyle =\frac{1}{10-1}\left[\left(227-222.10\right)^2+\left(222-222.10\right)^2+...+\left(221-222.10\right)^2\right]$

$\displaystyle =42.12$

The standard deviation is just the square root of the variance. That is

$\displaystyle s_{no\:aging}=\sqrt{s^2}=\sqrt{42.12}=6.49$

For the samples with AGING:

We know, based on our answer in Exercise 1.3, that the sample mean for samples with aging is $\displaystyle \overline{x}_{no\:aging}=209.90$.

To compute for the sample variance, we shall use the formula

$\displaystyle s^2=\sum _{i=1}^n\frac{\left(x_i-\overline{x}\right)^2}{n-1}$

The formula states that we need to get the sum of $\displaystyle \left(x_i-\overline{x}\right)^2$, so we can use a table to solve $\displaystyle \left(x_i-\overline{x}\right)^2$ for every sample.

Note: You can refer to the solution of Exercise 1.7 on how to use a table to solve for the variance.

The variance is

$\displaystyle \left(s^2\right)_{aging}=\sum _{i=1}^{10}\:\frac{\left(x_i-\overline{x}\right)^2}{n-1}$

$\displaystyle =\frac{1}{10-1}\left[\left(219-209.90\right)^2+\left(214-209.90\right)^2+...+\left(205-209.90\right)^2\right]$

$\displaystyle =23.62$

The standard deviation is just the square root of the variance. That is

$\displaystyle s_{aging}=\sqrt{s^2}=\sqrt{23.62}=4.86$

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