The average lifespan of a CleanFreek dishwasher is normally distributed with a mean of 8 years and a standard deviation of 1.6 years.

Standard Normal Distribution Table
a. What is the probability that a CleanFreek dishwasher will last longer than 10 years?
P(X > 10)=
 

b. What is the probability that a CleanFreek dishwasher will last fewer than 7 years?
P(X < 7)=
 

c. What length warranty should be established on the dishwashers so that no more than 1.8% of the units will need to be replaced under warranty?
x=
 
years
Round to 2 decimal places.

We are dealing with a normally distributed lifespan of CleanFreek dishwashers with:

- Mean (μ) = 8 years
- Standard deviation (σ) = 1.6 years

We can solve these using the standard normal distribution, converting the values into z-scores using the formula:
\[
z = \frac{X - \mu}{\sigma}
\]
Where:
- \(X\) is the value we are interested in,
- \( \mu \) is the mean,
- \( \sigma \) is the standard deviation.

Let's solve each part:

### a. What is the probability that a CleanFreek dishwasher will last longer than 10 years?
We need to find \(P(X > 10)\).

1. First, calculate the z-score for \(X = 10\):
   \[
   z = \frac{10 - 8}{1.6} = \frac{2}{1.6} = 1.25
   \]
2. Now, using the z-score table or a calculator, find the probability corresponding to \(z = 1.25\).
   The cumulative probability for \(z = 1.25\) is approximately 0.8944, meaning 89.44% of dishwashers will last less than 10 years.
   
   Therefore, the probability that a dishwasher will last longer than 10 years is:
   \[
   P(X > 10) = 1 - 0.8944 = 0.1056
   \]
   So, the probability is **0.1056** or **10.56%**.

### b. What is the probability that a CleanFreek dishwasher will last fewer than 7 years?
We need to find \(P(X < 7)\).

1. Calculate the z-score for \(X = 7\):
   \[
   z = \frac{7 - 8}{1.6} = \frac{-1}{1.6} = -0.625
   \]
2. Using the z-score table or a calculator, find the probability corresponding to \(z = -0.625\).
   The cumulative probability for \(z = -0.625\) is approximately 0.2660.
   
   Therefore, the probability that a dishwasher will last fewer than 7 years is:
   \[
   P(X < 7) = 0.2660
   \]
   So, the probability is **0.2660** or **26.60%**.

### c. What length warranty should be established on the dishwashers so that no more than 1.8% of the units will need to be replaced under warranty?

We need to find \(x\) such that \(P(X < x) = 0.018\).

1. From the standard normal distribution table, the z-score that corresponds to a cumulative probability of 0.018 is approximately \(z = -2.08\).
   
2. Using the z-score formula, we can solve for \(x\):
   \[
   z = \frac{x - 8}{1.6}
   \]
   Plugging in \(z = -2.08\):
   \[
   -2.08 = \frac{x - 8}{1.6}
   \]
   Solving for \(x\):
   \[
   x = 8 + (-2.08 \times 1.6) = 8 - 3.33 = 4.67
   \]

So, the warranty length should be **4.67 years** to ensure that no more than 1.8% of the dishwashers need replacement under warranty.

---

Do you want further details or have any questions? Here are 5 related questions you might explore next:

1. What is the probability that a CleanFreek dishwasher will last between 6 and 9 years?
2. How does changing the standard deviation affect the probability calculations?
3. What if the average lifespan was reduced to 7.5 years? How would the warranty change?
4. How can we calculate the median lifespan of these dishwashers?
5. What is the probability that a dishwasher lasts exactly 8 years?

**Tip**: When dealing with normally distributed variables, using z-scores simplifies finding probabilities in any normal distribution!

The average lifespan of a CleanFreek dishwasher is normally distributed with a mean of 8 years and a standard deviation of 1.6 years.

Standard Normal Distribution Table

a. What is the probability that a CleanFreek dishwasher will last longer than 10 years?
P(X > 10)=

b. What is the probability that a CleanFreek dishwasher will last fewer than 7 years?
P(X < 7)=

c. What length warranty should be established on the dishwashers so that no more than 1.8% of the units will need to be replaced under warranty?
x= years

Explore the probability of a CleanFreek dishwasher lasting more than 10 years or fewer than 7 years, and calculate the warranty length ensuring less than 1.8% replacements. Learn the use of z-scores and normal distribution with a mean of 8 years and a standard deviation of 1.6 years.

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