A used car dealership has found that the length of time before a major repair is required on the cars it sells is normally distributed with a mean of 10 months and a standard deviation of 3 months. If the dealer wants only 5 percent of the cars to fail before the end of the guarantee period for how many months should the cars be guaranteed?

You will need to find the z-score using a z-table. Remember the question is asking that the dealer wants only 5% of the cars to fail. Keep that in mind when looking for the z-score.

z = (x - mean)/sd -->z-score formula

Substitute the z-score you found, the mean, and the standard deviation into the formula, then solve for x.

I'll let you take it from there.

23

x=15 month

To determine how many months the cars should be guaranteed, we need to find the value that corresponds to the 5th percentile of the normal distribution.

Here's how we can calculate it:

Step 1: Find the Z-score
The Z-score represents the number of standard deviations a particular value is away from the mean. We can calculate the Z-score using the formula:

Z = (X - μ) / σ

Where:
Z = Z-score
X = Value we want to find
μ = Mean of the distribution (10 months)
σ = Standard deviation of the distribution (3 months)

Step 2: Find the corresponding percentile
The Z-score corresponds to a percentile in the standard normal distribution. We can use a Z-table or a calculator to find the percentile.

Step 3: Find the value
Once we have the percentile, we can convert it back to the original units (months) using the formula:

X = Z * σ + μ

Now, let's calculate it:

Step 1:
Z = (X - μ) / σ
5th percentile corresponds to Z = -1.645 (approximate value from the Z-table)

Step 2:
Using a Z-table or calculator, we can find that the 5th percentile corresponds to a Z-score of -1.645.

Step 3:
X = Z * σ + μ
X = -1.645 * 3 + 10
X ≈ 4.965

Therefore, the cars should be guaranteed for approximately 4.965 months to ensure that only 5 percent of the cars fail before the end of the guarantee period.