Question

The working lifetime, in years, of a particular model of bread maker is normally distributed with mean 10 and variance 4. Calculate the 12th percentile of the working lifetime, in years.

Answer 1

The 12th percentile of the working lifetime is 7.65 years.

Explanation:

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

In a set with mean
`\mu` and standard deviation(which is the square root of the variance)
`\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.

In this question, we have that:

`\mu = 10, \sigma = √(4) = 2`

12th percentile:

X when Z has a pvalue of 0.12. So X when Z = -1.175.

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

`-1.175 = (X - 10)/(2)`

`X - 10 = -1.175*2`

`X = 7.65`

The 12th percentile of the working lifetime is 7.65 years.