Question

for a sample of 110 transformers built for heavy industry, the mean and standard deviation of the number of sags per week were 363 and 21, respectively. also, the mean and standard deviation of the number of swells per week were 151 and 15, respectively. consider a transformer that has 285 sags and 174 swells in a week.
a. would you consider 285 sags per week unusual, statistically? explain.
select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. yes. the z - score is , meaning that this is an outlier and almost every other transformer has more sags.
(round to two decimal places as needed.)
b. yes. the z - score is , meaning that this is an outlier and almost every other transformer has fewer sags.
(round to two decimal places as needed.)
c. no. the z - score is , meaning that the number of sags is not unusual and is not an outlier.
(round to two decimal places as needed.)
d. no. the z - score is , meaning that less than approximately 68% of transformers have a number of sags closer to the mean.
(round to two decimal places as needed.)

Explanation:

Step1: Recall z - score formula

The z - score formula is $z=\frac{x-\mu}{\sigma}$, where $x$ is the value, $\mu$ is the mean and $\sigma$ is the standard deviation. Given $\mu = 363$, $\sigma=21$ and $x = 285$.

Step2: Calculate the z - score

Substitute the values into the formula: $z=\frac{285 - 363}{21}=\frac{- 78}{21}\approx - 3.71$.

Step3: Determine if it's unusual

A z - score with $|z|>2$ is often considered unusual. Since $| - 3.71|>2$, this value is an outlier.

Answer:

A. Yes. The z - score is - 3.71, meaning that this is an outlier and almost every other transformer has more sags.