Question

Voltage sags and swells. The power quality of a transformer is measured by the quality of the voltage. Two causes of poor power quality are "sags" and "swells." A sag is an unusual dip and a swell is an unusual increase in the voltage level of a transformer. The power quality of transformers built in Turkey was investigated in Electrical Engineering (Vol. 95, 2013). For a sample of 103 transformers built for heavy industry, the mean number of sags per week was 353 and the mean number of swells per week was 184. Assume the standard deviation of the sag distribution is 30 sags per week and the standard deviation of the swell distribution is 25 swells per week.

a. For a sag distribution with any shape, what proportion of transformers will have between 263 and 443 sags per week? Which rule did you apply and why?

b. For a sag distribution that is mound-shaped and symmetric, what proportion of transformers will have between 263 and 443 sags per week? Which rule did you apply and why?

c. For a swell distribution with any shape, what proportion of transformers will have between 109 and 259 swells per week? Which rule did you apply and why?

d. For a swell distribution that is mound-shaped and symmetric, what proportion of transformers will have between 109 and 259 swells per week? Which rule did you apply and why?

Short answer

a) 3

b) The distribution is symmetric as well as mound-shaped. The empirical rule is utilized here.

c) The transformer's z-score is 3 for 259 swells each week.

d) The transformers percentage with 109 to 259 swells every week is approximately 99.7% of the data within 3 standard deviations of the mean.

Step-by-step solution

(a) Proportion of transformers

Find the fraction of transformers that will experience 263 to 443 sags weekly.

The average number of sags each week in this area was 353. That is,$\mu =35.3,$and the sag distribution's standard error is 30 sags each week. In other words,$\sigma =30$

The z-score formula is as follows:

$z=\frac{x-\sigma }{\sigma }$

The z-score for x = 263 is as follows:

In the z-score calculation, substitute 263 as x, 353 as $\mu$, and 30 as in the formula of z-score.

As a result, the transformer's z-score will be 263 sags per week, which isformulae.

The z-score for x = 443 is obtained below.

Substitute 443 as x, 353 as $\sigma$, and 30 as $\sigma$in the z-score formula.

$$\begin{gathered} z=\frac{443-353}{30} \\ =\frac{90}{30} \\ =3 \end{gathered}$$

The transformer has a z-score of 3 and 443 sags each week.

Therefore, the transformer's z-score will be 263 sags per week, 3 standard deviations below the mean, and the transformer's z-score will be 443 sags per week, 3 standard deviations above the average.

At least $\frac{8}{9}$of the measures will fall within 3 standard deviations of the mean, according to Chebyshev's rule from Rule 2.1.

(b) Mound-shaped and symmetric proportion of transformers

The distribution is symmetric as well as mound-shaped. The empirical rule is utilized here. Using Rule 2.2's Empirical rule, the fraction of transformers with sags between 263 and 443 every week is "roughly 99.7% of the measurements will fall within 3 standard deviations from the mean."

(c) Swell distribution with any shape as well as the proportion of transformers

Determine the fraction of transformers that will have 109 to 259 swells every week.

The average number of swells each week was 184 in this area. That is, $\mu =184$and the swell distribution's standard deviation is 25 swells each week. This equal $\sigma =25$

The z-score for $x=109$is calculated as follows.

Substitute 109 as x, 184 as $\mu$, and 25 as $\sigma$in the z-score formula.

$$\begin{gathered} \text{z}=\frac{109-184}{25} \\ =\frac{-75}{25} \\ =-3 \end{gathered}$$

The transformer's z-score will be 109 swells each week is -3.

The z-score for x =259 is obtained below

Substitute 259 as x, 184 as $\mu$.and 25 as $\sigma$in the z-score formula

$$\begin{gathered} z=\frac{259-184}{25} \\ =\frac{75}{25} \\ =3 \end{gathered}$$

The transformer's z-score is 3 for 259 swells each week.

Thus, the transformer's z-score will be 109 swells per week, 3 standard deviations below the mean, and the transformer's z-score will be 259 swells per week, which is 3 standard deviations above the average.

Using Rule 2.1's Chebyshev's rule, "At least $\frac{8}{9}$one of the measures will be within 3 inches the mean, standard deviation."

(d) Mound-shaped and symmetric proportion of transformers as well as applied rule

The distribution is symmetrical as well as mound-shaped. The empirical rule is utilized here. Using the Empirical rule from Rule 2.2, the percentage of transformers with 109 to 259 swells every week is roughly 99.7% of the data falling within 3 of the mean, standard deviation.