Suppose a tire manufacturer wants to set a mileage guarantee on its new XB 70 tire. Life tests revealed that the mean mileage is 47,900 miles and the standard deviation is 2,050. The manufacturer wants to set the guaranteed mileage so that no more than five percent of the tires will have to be replaced. What guaranteed mileage should the manufacturer set? (assume normality)

The worst 5% of the tires (lowest mileage) will have a Z-score of -1.645. Get that from a normal distribution tabe or calculator, such as

http://www.measuringusability.com/zcalcp.php

Subtract 1.645*2050 from the mean mileage for your answer.

Suppose a tire manufacturer wants to set a mileage guarantee on its new XB 70 tire. Life tests revealed that the mean mileage is 47,900 miles and the standard deviation is 2,050. The manufacturer wants to set the guaranteed mileage so that no more than five percent of the tires will have to be replaced. What guaranteed mileage should the manufacturer set? (assume normality

To determine the guaranteed mileage, we need to find the mileage value that corresponds to the fifth percentile (5% of the tires) in a normal distribution with a mean of 47,900 miles and a standard deviation of 2,050.

Here's a step-by-step approach to finding the guaranteed mileage:

Step 1: Convert the desired percentile (5%) to a z-score.
The z-score is the number of standard deviations away from the mean a particular observation falls. To convert the percentile to a z-score, we use the inverse standard normal distribution table (also known as the Z-table) or a calculator. The fifth percentile corresponds to a z-score of approximately -1.645.

Step 2: Use the z-score to find the corresponding raw value (mileage).
The z-score is calculated using the formula z = (X - μ) / σ, where X is the raw value, μ is the mean, and σ is the standard deviation. Rearranging the formula, we have X = z * σ + μ.

Substituting the values, X = (-1.645) * (2,050) + (47,900).

Step 3: Calculate the guaranteed mileage.
Compute the value of X from the equation above:

X = (-1.645) * (2,050) + (47,900)
X ≈ -3,371 + 47,900
X ≈ 44,529

Therefore, the manufacturer should set the guaranteed mileage for the new XB 70 tire at approximately 44,529 miles to ensure that no more than 5% of the tires will need to be replaced.

To calculate the guaranteed mileage, we need to find the value that corresponds to the 5th percentile (or the point below which 5% of the data falls) of a normal distribution.

Here are the steps to find the guaranteed mileage:

Step 1: Identify the parameters of the distribution:
Mean (μ) = 47,900 miles
Standard Deviation (σ) = 2,050 miles

Step 2: Determine the z-score corresponding to the desired percentile:
Since the manufacturer wants to set the mileage guarantee so that no more than 5% of the tires will have to be replaced, we need to find the z-score that corresponds to the 5th percentile.

Step 3: Use a standard normal distribution table or statistical software to find the z-score:
Using a standard normal distribution table or statistical software, we find that the z-score corresponding to the 5th percentile is approximately -1.645.

Step 4: Use the z-score formula to find the guaranteed mileage:
To find the guaranteed mileage, we can use the formula:
X = μ + (z * σ)
where X is the guaranteed mileage, μ is the mean, z is the z-score, and σ is the standard deviation.

Using the values we have:
X = 47,900 + (-1.645 * 2,050)
X ≈ 47,900 - 3,370.25
X ≈ 44,529.75

Therefore, the manufacturer should set the guaranteed mileage at approximately 44,530 miles to ensure that no more than five percent of the tires will have to be replaced.