Question

A flashlight battery manufacturer makes a model of battery whose mean shelf life is three years and four months, with a standard deviation of three months. The distribution is approximately normal. One production run of batteries in the factory was 25,000 batteries. How many of those batteries can be expected to last between three years and one month and three years and seven months?

Answer 1

Answer:

17,065 of those batteries can be expected to last between three years and one month and three years and seven months

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

In a set with mean $\mu$ and standard deviation $\sigma$, 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 problem:

I will calculate the time in months. Each year has twelve months.

Mean shelf life is three years and four months, with a standard deviation of three months. So

$\mu = 3*12 + 4 = 40$

$\sigma = 3$

Proportion lasting between three years and one month and three years and seven months:

This is the pvalue of Z when X = 3*12 + 7 = 43 subtracted by the pvalue of Z when X = 3*12 + 1 = 37

X = 43

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

$Z = \frac{43 - 40}{3}$

$Z = 1$

$Z = 1$ has a pvalue of 0.8413.

X = 37

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

$Z = \frac{37 - 40}{3}$

$Z = -1$

$Z = -1$ has a pvalue of 0.1587.

0.8413 - 0.1587 = 0.6826

Out of 25,000 batteries:

68.26% of the batteries are expected to last between three years and one month and three years and seven months.

0.6826*25000 = 17,065

17,065 of those batteries can be expected to last between three years and one month and three years and seven months