Question

A battery has a lifetime that is approximately normally distributed with a mean of 600 hours and a standard deviation of 50 hours. Suppose you purchased one of these batteries and it only lasted for 400 hours. Which of the following conclusions is the most appropriate given this information?

A. Your battery is likely defective since such poor performance is extremely unlikely.
B. Your battery is definitely below average since it is below the mean.
C. There is a 50-50 chance of a battery having a worse than average performance, so the performance of your battery is not surprising.

Answer 1

A. Your battery is likely defective since such poor performance is extremely unlikely.

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

If X is more than two standard deviations from the mean, it is considered an unlikely outcome.

In this question, we have that:

`\mu = 600, \sigma = 50`

Which of the following conclusions is the most appropriate given this information?

Lasted 400 hours, so X = 400.

`Z = (X - \mu)/(\sigma)`

`Z = (400 - 600)/(50)`

`Z = -4`

4 standard deviations from the mean, so unlikely.

So the correct answer is:

A. Your battery is likely defective since such poor performance is extremely unlikely.