Your company manufactures hot water heaters. The life spans of the products are known to be normally distributed with a mean of 13 years and a standard deviation of 1.5 years.

Question

1. What is the probability that a random selected hot water heater has a life span of between 12 and 15 years?
2. What is the probability that a random selected hot water heater has a life span of more than 15 years?
3. You are buying a new hot water heater, based on the information in the problem, between how many years would you expect your hot water heater to last. (Give a range of numbers). Explain your answer. 
4. What is the probability that the mean life span in a group of 10 randomly selected hot water heaters is between 12 and 15 years?
5. You want to set the warranty on your product so that you have to replace at most 5% of the hot water heaters that you sell. How many years should you claim on your warranty? (Round answer to nearest whole number of years.) Explain your answer. 

Answer 1

µ = 13

σ = 1.5

1. normalcdf(lower: 12, upper: 15, µ: 13, σ: 1.5) ≈ 0.656

2. normalcdf(lower: 15, upper: 999, µ: 13, σ: 1.5) ≈ 0.091

3. By the Empirical Rule, we expect the water heater to last, at most, two standard deviations less or greater than the mean.

13 + 2(1.5) = 16

13 - 2(1.5) = 10

I expect my water heater to last between 10 and 16 years.

4. µ = 13

σ = σ/√n = 1.5 / √10 ≈ 0.47434165

normalcdf(lower: 12, upper: 15, µ: 13, σ: 0.47434165) ≈ 0.982

5. invNorm(area: 0.05, µ: 13, σ: 1.5, Tail: LEFT) ≈ 11 years

The warranty on each water heater should be set to 11 years because we expect 5% of water heaters to last no longer than 10.533 years.