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

Answer:

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

Step-by-step 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:

$Z = \frac{X - \mu}{\sigma}$

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:

$\mu = 10, \sigma = \sqrt{4} = 2$

12th percentile:

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

$Z = \frac{X - \mu}{\sigma}$

$-1.175 = \frac{X - 10}{2}$

$X - 10 = -1.175*2$

$X = 7.65$

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