Let's break down the question:

1. **Given dataLet's break down the question:

2. **Given **

   • The replacement times are normally distributed.
   • Mean $\mu = 11.6$ years.
   • Standard deviation $\sigma = 1.3$ years.

   First Part: Find the probability that a randomly selected quartz time piece has a replacement time less than 8 years.

   This is asking for $P(X < 8)$, where $X$ is the normal random variable representing the replacement time.

   To solve this:

   • We need to calculate the Z-score using the formula:

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

   For $X = 8$:

   $Z = \frac{8 - 11.6}{1.3} = \frac{-3.6}{1.3} \approx -2.769$

   Now, we can look up the Z-score in the standard normal distribution table, or use a calculator. The Z-score of $-2.769$ corresponds to a cumulative probability of approximately 0.0028.

   So, the probability that a replacement time is less than 8 years is about 0.0028.

   Second Part: Warranty Calculation

   The company wants to set the warranty such that only 2.9% (or 0.029) of quartz timepieces will be replaced before the warranty expires. This is the area to the left of the desired warranty time in the normal distribution.

   • First, find the Z-score that corresponds to the 2.9% left-tail probability. Using a Z-table or calculator, the Z-score for 0.029 is approximately $-1.88$.

   • Now use the Z-score formula in reverse to find the corresponding warranty time, $X$:

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

   Solving for $X$:

   $X = Z \cdot \sigma + \mu$

   Plugging in the values:

   $X = (-1.88) \cdot 1.3 + 11.6 \approx -2.444 + 11.6 = 9.156$

   So, the warranty should be approximately 9.2 years (rounding to 1 decimal place).

Let me know if you'd like further explanation or have more questions!

Here are five related questions you might find useful:

1. How do we interpret a Z-score in the context of a normal distribution?
2. What are some common Z-scores and their corresponding probabilities?
3. How does changing the standard deviation affect the probability in a normal distribution?
4. How can you determine the warranty period for different replacement percentages?
5. What are some methods to approximate probabilities when Z-tables are not available?

Tip: Always remember to check whether the normal distribution is appropriate for the data by considering skewness or kurtosis before applying Z-scores.