the typical saturday the oil change facility will perform 40 oil changes between 10 am and 12pm. treating this as a random sample, what mean oil change time would there be a 10% chance of being at or below? There is a 10% chance of being at or below a mean oil change time of ____ minutes
To determine the mean oil change time that would correspond to a 10% chance of being at or below, we need to use statistical concepts and techniques. Here are the steps to calculate it:
1. Firstly, we need to gather some information:
- The number of oil changes performed (n) = 40
- The time range for the oil changes (in this case, between 10 am and 12 pm)
2. Since we have a time range, we can assume that the distribution of oil change times is approximately normal.
3. Next, we need to find the z-score corresponding to a 10% chance of being at or below a certain value. To do this, we can use a standard normal distribution table or a calculator.
4. The z-score corresponding to a 10% chance is -1.28 (approximately). This means that 10% of values fall below -1.28 standard deviations from the mean.
5. To find the mean oil change time corresponding to the 10th percentile (10% chance), we can use the formula:
mean + (z * standard deviation) = X
where X is the oil change time.
6. However, we don't have the standard deviation of oil change times. To estimate it, we can use the Central Limit Theorem. Since the number of oil changes performed is reasonably large (n = 40), we can assume that the sample mean follows a normal distribution.
7. Instead of the population standard deviation, we can use the sample standard deviation (s) as an estimate for the population standard deviation.
8. If the facility has historical data on oil change times, we can calculate the sample standard deviation (s) using the formula:
s = √((Σ(xi - x̄)^2) / (n - 1))
where xi represents each individual oil change time, and x̄ is the sample mean.
9. After estimating the standard deviation, we can plug the values into the formula and solve for the mean:
mean + (-1.28 * standard deviation) = X
By following these steps and calculating the necessary values, you should be able to determine the mean oil change time that corresponds to a 10% chance of being at or below.