Question

The average lifespan of a CleanFreek dishwasher is normally distributed with a mean of 9 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 11 years?
P(X > 11)=P(X > 11)=
b. What is the probability that a CleanFreek dishwasher will last fewer than 8 years?
P(X < 8)=P(X < 8)=
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?

Answer 1

a. The probability that a CleanFreek dishwasher will last longer than 11 years is approximately 0.8944. b. The probability that a CleanFreek dishwasher will last fewer than 8 years is approximately 0.2659. c. A warranty of approximately 12.3 years should be established to ensure that no more than 1.8% of the units will need to be replaced under warranty.

Step-by-step explanation:

a. To find the probability that a CleanFreek dishwasher will last longer than 11 years, we need to calculate the cumulative probability of the z-score of 11. First, we calculate the z-score using the formula:

z = (x - mean) / standard deviation.

Substituting the values, we get z = (11 - 9) / 1.6 = 1.25.

Using a standard normal distribution table, we find that the cumulative probability for a z-score of 1.25 is approximately 0.8944.

Therefore, the probability that a CleanFreek dishwasher will last longer than 11 years is 0.8944.

b. Similarly, to find the probability that a CleanFreek dishwasher will last fewer than 8 years, we calculate the z-score using the formula: z = (x - mean) / standard deviation.

Substituting the values, we get z = (8 - 9) / 1.6 = -0.625.

Using a standard normal distribution table, we find that the cumulative probability for a z-score of -0.625 is approximately 0.2659. Therefore, the probability that a CleanFreek dishwasher will last fewer than 8 years is 0.2659.

c. To find the length of warranty that 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 calculate the z-score corresponding to a cumulative probability of 1 - 0.018 = 0.982.

Using a standard normal distribution table, we find that the z-score for a cumulative probability of 0.982 is approximately 2.06.

Then, we can use the formula z = (x - mean) / standard deviation to solve for x, the length of warranty. Rearranging the formula, we get x = mean + z * standard deviation = 9 + 2.06 * 1.6 = 12.296 years.

Therefore, a warranty of approximately 12.3 years should be established to ensure that no more than 1.8% of the units will need to be replaced under warranty.