Mathematics, 08.04.2020 00:09 stonerbabyy

The operation manager at a tire manufacturing company believes that the mean mileage of a tire is 34,641 miles, with a standard deviation of 2480 miles. 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

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Answer from: Brock2002

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

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 $\mu$ and standard deviation $\sigma$ , the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

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:

$\mu = 34631, \sigma = 2480, n = 121, s = \frac{2480}{\sqrt{121}} = 225.45$

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

$Z = \frac{X - \mu}{\sigma}$

By the Central Limit Theorem

$Z = \frac{X - \mu}{s}$

$Z = \frac{34985 - 34631}{225.45}$

Z = 1.525

Z = 1.525 has a pvalue of 0.93635

X = 34297

$Z = \frac{X - \mu}{s}$

$Z = \frac{34297 - 34631}{225.45}$

Z = -1.525

Z = -1.525 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

Answer from: Quest

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Answer from: Quest

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