Respuesta :
Answer:
0.4938 = 49.38% probability that the sample mean would differ from the population mean by less than 170 miles in a sample of 211 tires if the manager is correct
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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 [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
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:
[tex]\mu = 36309, \sigma = 4693, n = 211, s = \frac{4693}{\sqrt{211}} = 323.1[/tex]
What is the probability that the sample mean would differ from the population mean by less than 170 miles in a sample of 211 tires if the manager is correct?
This is the pvalue of Z when X = 36309 + 170 = 36479 subtracted by the pvalue of Z when X = 36309 - 170 = 36139. So
X = 36479
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{36479 - 36309}{323.1}[/tex]
[tex]Z = 0.53[/tex]
[tex]Z = 0.53[/tex] has a pvalue of 0.7019
X = 36139
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{36139 - 36309}{323.1}[/tex]
[tex]Z = -0.53[/tex]
[tex]Z = -0.53[/tex] has a pvalue of 0.2981
0.7019 - 0.2081 = 0.4938
0.4938 = 49.38% probability that the sample mean would differ from the population mean by less than 170 miles in a sample of 211 tires if the manager is correct
