Answer:
For a mean oil change time of 20.51 minutes there would be a 10% chance of being at or below
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
Problems of normally distributed samples are solved using the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean
and standard deviation
, the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean
and standard deviation
.
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
In this problem, we have that:
![\mu = 21.3, \sigma = 3.9, n = 40, s = \frac{3.9}{\sqrt{40}} = 0.6166](https://tex.z-dn.net/?f=%5Cmu%20%3D%2021.3%2C%20%5Csigma%20%3D%203.9%2C%20n%20%3D%2040%2C%20s%20%3D%20%5Cfrac%7B3.9%7D%7B%5Csqrt%7B40%7D%7D%20%3D%200.6166)
Treating this as a random sample, at what mean oil-change time would there be a 10% chance of being at or below?
This is the 10th percentile, which is X when Z has a pvalue of 0.1. So it is X when Z = -1.28.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
By the Central Limit Theorem
![Z = \frac{X - \mu}{s}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7Bs%7D)
![-1.28 = \frac{X - 21.3}{0.6166}](https://tex.z-dn.net/?f=-1.28%20%3D%20%5Cfrac%7BX%20-%2021.3%7D%7B0.6166%7D)
![X - 21.3 = -1.28*0.6166](https://tex.z-dn.net/?f=X%20-%2021.3%20%3D%20-1.28%2A0.6166)
![X = 20.51](https://tex.z-dn.net/?f=X%20%3D%2020.51)
For a mean oil change time of 20.51 minutes there would be a 10% chance of being at or below