Question

Voltage sags and swells. Refer to the Electrical Engineering (Vol. 95, 2013) study of transformer voltage sags and swells, Exercise 2.76 (p. 110). Recall that 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. Suppose one of the transformers is randomly selected and found to have 400 sags and 100 swells in a week.

a. Find the z-score for the number of sags for this transformer. Interpret this value.

b. Find the z-score for the number of swells for this transformer. Interpret this value.

Short answer

1. 1.567
2. -3.36

Step-by-step solution

Calculating the z-score for the total number of sags for the transformer

The calculation of the z-score for the sags is shown below:

$$\begin{gathered} \mathbf{\text{z - score =}}\frac{\left(\text{Observed} \text{value - Mean} \text{number} \text{of} \text{sags}\right)}{\mathbf{\text{Standard}}\mathbf{ }\mathbf{\text{deviation}}} \\ \mathbf{\text{=}}\frac{\left(\text{400 - 353}\right)}{\mathbf{\text{30}}} \\ \mathbf{\text{= 1.567}} \end{gathered}$$

The z-score from the table is found to be 0.94, which means around 94 percent of the transformers have less number of sags.

Computing the z-score for the total number of swells for the transformer

The calculation of the z-score for the swells is shown below:

$$\begin{gathered} \mathbf{\text{z - score =}}\frac{\left(\text{Observed value - Mean} \text{number} \text{of} \text{swells}\right)}{\mathbf{\text{Standard}}\mathbf{ }\mathbf{\text{deviation}}} \\ \mathbf{\text{=}}\frac{\left(\text{100 - 184}\right)}{\mathbf{\text{25}}} \\ \mathbf{\text{= - 3.36}} \end{gathered}$$

The z-score from the table is found to be 0.0004, which means around 4 percent of the transformers have fewer swells.