Answer:
87.27% probability that the sample mean would differ from the population mean by less than 344 miles in a sample of 121 tires if the manager is correct
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 = (X - \mu)/(\sigma)](https://img.qammunity.org/2021/formulas/mathematics/college/c62rrp8olhnzeelpux1qvr89ehugd6fm1f.png)
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 = 34631, \sigma = 2480, n = 121, s = (2480)/(√(121)) = 225.45](https://img.qammunity.org/2021/formulas/mathematics/college/vxncwesxpdy91vccoz3m35owqh56bdb0oi.png)
What is the probability that the sample mean would differ from the population mean by less than 344 miles in a sample of 121 tires if the manager is correct
This is the pvalue of Z when X = 34641 + 344 = 34985 subtracted by the pvalue of Z when X = 34641 - 344 = 34297. So
X = 34985
![Z = (X - \mu)/(\sigma)](https://img.qammunity.org/2021/formulas/mathematics/college/c62rrp8olhnzeelpux1qvr89ehugd6fm1f.png)
By the Central Limit Theorem
![Z = (X - \mu)/(s)](https://img.qammunity.org/2021/formulas/mathematics/college/qbjdi63swemoz9mdzfqtue91aagng8mdqs.png)
![Z = (34985 - 34631)/(225.45)](https://img.qammunity.org/2021/formulas/mathematics/college/jhwabwydbf3qci2u3se0gd780731j1ghll.png)
![Z = 1.525](https://img.qammunity.org/2021/formulas/mathematics/college/pw3jpigtlspnidpfowrat2wgi9emm6jivz.png)
has a pvalue of 0.93635
X = 34297
![Z = (X - \mu)/(s)](https://img.qammunity.org/2021/formulas/mathematics/college/qbjdi63swemoz9mdzfqtue91aagng8mdqs.png)
![Z = (34297 - 34631)/(225.45)](https://img.qammunity.org/2021/formulas/mathematics/college/6s9q5zbhghimo0cehyi5z70bft94oa3elb.png)
![Z = -1.525](https://img.qammunity.org/2021/formulas/mathematics/college/yh0zfm459ew1na15rsbmyrrg6qaa59z6d1.png)
has a pvalue of 0.06365
0.93635 - 0.06365 = 0.8727
87.27% probability that the sample mean would differ from the population mean by less than 344 miles in a sample of 121 tires if the manager is correct