A certain company makes 12-volt car batteries. After many years of product testing, the company knows that the average life of a battery is normally distributed, with a mean of 45 months and a standard deviation of 7 months. If the company does not want to make refunds for more than 10% of its batteries under the full-refund guarantee policy, for how long should the company guarantee the batteries (to the nearest month)?

The choice of answers:
58 months
53 months
36 months
45 months
49 months

*I came out with 45 but was told that was wrong. What did I do wrong. Thanks to those who answer in advance.

45 months = mean! In a normal distribution, mean = median, 50% are above and are below the mean. How did you "come out with 45"?

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find Z score related to that proportion.

Z = 1.28

Z = (score-mean)/SD

1.28 = (score-45)/7

Solve the above equation.

To solve this problem, we need to find the length of time the company should guarantee the batteries so that 10% or fewer will fail during the guarantee period. Let's break down the steps to solve this problem:

1. Define the problem:
We need to determine the length of time for the guarantee period.

2. Understand the problem:
The company wants to limit the number of refunds for more than 10% of its batteries. In other words, they want to ensure that no more than 10% of the batteries fail during the guarantee period.

3. Identify the relevant values from the problem:
Mean (μ) = 45 months
Standard deviation (σ) = 7 months
Desired proportion of success: p = 1 - 10% = 0.90

4. Determine the z-score:
To find the z-score, we need to use the formula:
z = (x - μ) / σ

Rearranging the formula, we get:
x = (z * σ) + μ

5. Find the z-value for the desired proportion:
To find the z-value corresponding to a success proportion of 0.90, we consult a standard normal distribution table or use a calculator. The z-value for a proportion of 0.90 is approximately 1.28.

6. Calculate the guarantee period:
Using the formula from step 4, we substitute the values into the equation:
x = (1.28 * 7) + 45
x ≈ 53.96

7. Round the guarantee period:
Since we cannot have partial months, we need to round the answer. So, the company should guarantee the batteries for approximately 54 months.

Therefore, the correct answer from the given choices is 53 months.

Note: It's important to pay attention to the rounding instructions provided in the question. The answer may have been marked incorrect if you rounded to 45 instead of 54.