Question

company xyz know that replacement times for the quartz time pieces it produces are normally distributed with a mean of 14.3 years and a standard deviation of 1.6 years.
find the probability that a randomly selected quartz time piece will have a replacement time less than 9.2 years.
$p(x < 9.2\text{ years}) = \square$
enter your answer accurate to 4 decimal places. answers obtained using exact z - scores or z - scores rounded to 3 decimal places are accepted.

Explanation:

Step1: Calculate the z - score

The formula for the z - score is \(z=\frac{x-\mu}{\sigma}\), where \(x = 9.2\), \(\mu=14.3\), and \(\sigma = 1.6\).

$$z=\frac{9.2 - 14.3}{1.6}=\frac{- 5.1}{1.6}=-3.1875$$

Step2: Find the probability using the standard normal distribution

We want to find \(P(X\lt9.2)\), which is equivalent to \(P(Z\lt - 3.1875)\).
Using a standard normal table (or a calculator with a normal - distribution function, e.g., in Excel: NORM.S.DIST(-3.1875, TRUE)), we get the probability.

Answer:

\(0.0007\)