Question

It has been observed that electrical connectors manufactured by Jolt Electrical Supply Company last an average of 18.2 months and follow a normal distribution with a standard deviation of 1.7 months. Jolt agrees to replace any connector that fails within 19 months. Out of 500 connectors sold, how many does Jolt expect to replace, on average? Place your answer, rounded to the nearest whole number, in the blank. For example, 123 would be a legitimate answer.

Answer 1

340

Explanation:

The first step is finding the percentage of those who fail within 19 months.

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 = (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, we have that:

`\mu = 18.2, \sigma = 1.7`

The proportion is the pvalue of Z when X = 19. So

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

`Z = (19 - 18.2)/(1.7)`

`Z = 0.47`

`Z = 0.47` has a pvalue of 0.6808.

Jolt is expected to replace 0.6808 of the connectors sold. Out of 500, that is, on average, 0.6808*500 = 340.

Jolt expects to replace, on average, 340 of the connectors sold.