Voltage sags and swells. Refer to the Electrical Engineering (Vol. 95, 2013) study of the power quality (sags and swells) of a transformer, Exercise 2.76 (p. 110). For transformers built for heavy industry, the distribution of the number of sags per week has a mean of 353 with a standard deviation of 30. Of interest is , that the sample means the number of sags per week for a random sample of 45 transformers.

a. Find$\mathbf{E}\left(\overline{\mathbf{\chi }}\right)$ and interpret its value.

b. Find$\mathbf{Var}\left(\overline{\mathbf{\chi }}\right).$

c. Describe the shape of the sampling distribution of$\overline{\mathbf{\chi }}.$

d. How likely is it to observe a sample mean a number of sags per week that exceeds 400?

The calculation is given below:

To find $\mathrm{E}\left(\overline{\mathrm{\chi }}\right)$

The weekly average value of sags is $\mu =353$ as well as the random sample is 45, which is

When a random sampling of n observations is chosen from a standard normal distribution, the sample distribution's mean is equal to the population distribution's mean, demonstrating that the population is normal.

Therefore,

$\begin{array}{c}{\mu }_{\overline{\mathrm{\chi }}}=\mu \\ =353\end{array}$

The value of localid="1652096362826" $\mathrm{E}\left(\overline{\mathrm{\chi }}\right)\mathrm{is}353$

The calculation is given below:

To find $\mathrm{Var}\left(\overline{\mathrm{\chi }}\right)$

The standard deviation of the weekly number of sags is $\sigma =30$

The standard deviation of the sampling distribution is ${\sigma }_{\overline{\mathrm{\chi }}}=\frac{\mathrm{\sigma }}{\sqrt{\mathrm{n}}}$

Therefore,

$\begin{array}{c}{\sigma }_{\overline{\mathrm{\chi }}}=\frac{30}{\sqrt{45}}\\ =4.4721\end{array}$

The variance is:

$\begin{array}{c}\mathrm{Var}\left(\overline{\mathrm{\chi }}\right)={\sigma }_{\overline{\mathrm{\chi }}}^2\\ ={\left(4.4721\right)}^2\\ =19.9996\\ \approx 20\end{array}$

Therefore, the value of $\mathrm{Var}\left(\overline{\mathrm{\chi }}\right)\mathrm{is}20$

The sampling distribution will be essentially normal for relatively significant sample sizes. Furthermore, the distribution has an asymmetric form.

Calculate the value of z at $\overline{\mathrm{\chi }}=400$

$\begin{array}{c}z=\frac{\overline{\mathrm{\chi }}-{\mu }_{\overline{\mathrm{\chi }}}}{{\sigma }_{\overline{\mathrm{\chi }}}}\\ =\frac{400-353}{19.9996}\\ =\frac{47}{19.9996}\\ =2.35\end{array}$

The value of z is 2.35

Determine the probability that the sample mean count of sags each week is more than 400.