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.

Question

Mathematics

Cantu

30 September, 04:46

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.

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Hailie Klein · Answer 1

The percent of the pairs that last longer than six months-that is, 180 days is 95.637%

Step-by-step explanation:

Mean = xbar = 208 days

Standard deviation = σ = 14

The standardized score for 180 days is the value minus the mean then divided by the standard deviation.

z = (x - xbar) / σ = (180 - 208) / 14 = - 1.71

To determine the percent of boots that last longer than 180 days, we need this probability, P (x > 180) = P (z > (-1.71))

We'll use data from the normal probability table for these probabilities

P (x > 180) = P (z > (-1.71)) = 1 - P (z ≤ (-1.71)) = 1 - 0.04363 = 0.95637 = 95.637%