Please help asap! A tire company is considering a tire replacement company. They have agreed to replace any tire that fails with a life in the bottom 10% of the population. For a given population of tires, the life is a normal distribution with a mean of 27 months and a standard deviation of 3.8 months. What will be the maximum life of a tire to qualify for the replacement program?
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the Z score related to that proportion.
Z = (score - mean)/SD
Substitute the values and solve for the score (months).
To determine the maximum life of a tire that qualifies for the replacement program, we need to find the value that corresponds to the bottom 10% of the population.
1. Start by finding the Z-score corresponding to the bottom 10%.
- The area to the left of any given Z-score represents the percentage of data that falls below that value.
- Since we are interested in the bottom 10%, we need to find the Z-score that corresponds to the area of 0.10.
2. Use a Z-table or a statistical calculator to find the Z-score.
- A Z-table provides the percentage of data that falls below a given Z-score.
- In this case, we need to find the Z-score that corresponds to an area of 0.10.
3. Once you have the Z-score, use it to calculate the maximum life of a tire.
- The Z-score formula is: Z = (X - μ) / σ, where X is the tire life, μ is the mean, and σ is the standard deviation.
- Rearrange the formula to solve for X: X = Z * σ + μ.
- Substitute the Z-score and the given values of μ and σ to find the maximum tire life.
Let's go through the steps to find the maximum life of a tire that qualifies for the replacement program.
1. Finding the Z-score:
- From the given information, we need to find the Z-score that corresponds to an area of 0.10.
- Using a Z-table or a statistical calculator, locate the value closest to 0.10 in the table or enter 0.10 into the calculator.
- Find the corresponding Z-score. Let's say it is -1.28 (rounded to two decimal places).
2. Calculating the maximum tire life:
- Using the formula X = Z * σ + μ, substitute the values: X = -1.28 * 3.8 + 27.
- Calculate the expression: X = -4.864 + 27.
- The maximum life of a tire to qualify for the replacement program is approximately 22.14 months (rounded to two decimal places).
Therefore, any tire with a life of less than or equal to approximately 22.14 months would qualify for the replacement program.