You work in marketing for a company that produces work boots. Quality control has sent you a memo detailing the length of time before the boots wear out under heavy use. They find that the boots wear out in an average of 208 days, but the exact amount of time varies, following a normal distribution with a standard deviation of 14 days. For an upcoming ad campaign, you need to know the percent of the pairs that last longer than six months-that is, 180 days. Use the empirical rule to approximate this percent.
A) 2.5%
B) 5%
C) 95%
D) 97.5%

Question

Quizplus · Accepted Answer

The empirical rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Since 180 days is more than one standard deviation (14 days) below the mean (208 days), the percentage of boots lasting longer than 180 days would be in the upper tail, beyond one standard deviation from the mean, which corresponds to roughly 50% (half below the mean) plus half of the 68% (one standard deviation), totaling approximately 84%. However, the empirical rule doesn't directly give us the exact percentage for this calculation, but it indicates that a significant majority of the boots last longer than 180 days. The correct answer, based on the options provided and understanding of the empirical rule, would be that more than 84% but less than 97.5% of the boots last longer than 180 days, making D (97.5%) the closest choice, even though the empirical rule doesn't precisely calculate this specific percentage. The correct calculation involves using the Z-score and looking up the corresponding percentile in a standard normal distribution table, which wasn't done here. My explanation mistakenly suggested choosing an option based on an incorrect application of the empirical rule specifics.

You Work in Marketing for a Company That Produces Work