Company XYZ know that replacement times for the quartz time pieces it produces are normally distributed with a mean of 11.4 years and a standard deviation of 0.8 years. Find the probability that a randomly selected quartz time piece will have a replacement time less than 8.8 years? \( P(X<8.8 \) years \( )= \) Enter your answer accurate to 4 decimal places. Answers obtained using exact \( z \)-scores or \( z \)-scores rounded to 3 decimal places are accepted.

To find \( P(X < 8.8) \), we first need to calculate the \( z \)-score using the formula: \[ z = \frac{(X - \mu)}{\sigma} \] Where: - \( X = 8.8 \) years - \( \mu = 11.4 \) years (mean) - \( \sigma = 0.8 \) years (standard deviation) Plugging in the values: \[ z = \frac{(8.8 - 11.4)}{0.8} = \frac{-2.6}{0.8} = -3.25 \] Next, we look up the \( z \)-score of -3.25 in the standard normal distribution table or use a calculator. The cumulative probability for \( z = -3.25 \) is approximately 0.0006. Thus, \[ P(X < 8.8 \text{ years}) \approx 0.0006 \] Rounded to four decimal places, the probability is: **0.0006**