Question

The toasters produced by a company have a normally distributed life span with a mean of 5.8 years and a standard deviation of 0.9 years, what warranty should be provided so that the company is replacing at most 10% of their toasters sold? a. 4.3 years b. 5.9 years c. 4.5 years d. 4.6 years

Answer 1

Answer:

d. 4.6 years

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, we have that:

$\mu = 5.8, \sigma = 0.9$

What warranty should be provided so that the company is replacing at most 10% of their toasters sold?

Only those on the 10th percentile or lower will be replaced.

So the warranty is the value of X when Z has a pvalue of 0.10.

So it is X when $Z = -1.28$

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

$-1.28 = \frac{X - 5.8}{0.9}$

$X - 5.8 = -1.28*0.9$

$X = 4.6$

So the correct answer is:

d. 4.6 years