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## An investigator compares the durability of two different compounds used in the manufacture of a certain automobile brake lining. A sample of

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An investigator compares the durability of two different compounds used in the manufacture of a certain automobile brake lining. A sample of 243 brakes using Compound 1 yields an average brake life of 37,866 miles. A sample of 268 brakes using Compound 2 yields an average brake life of 45,789 miles. Assume that the population standard deviation for Compound 1 is 4414 miles, while the population standard deviation for Compound 2 is 2368 miles. Determine the 95% confidence interval for the true difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2. Step 2 of 3 : Calculate the margin of error of a confidence interval for the difference between the two population means. Round your answer to six decimal places.

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2021-08-28T11:03:49+00:00
2021-08-28T11:03:49+00:00 1 Answers
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## Answers ( )

Answer:The margin of error is of 623.2016.

The 95% confidence interval for the true difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2 is (-8546.2016, -7299.7984).

Step-by-step explanation:Before building the confidence interval, we need to understand the central limit theorem, and subtraction between normal variables.

Central Limit TheoremThe 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.

Subtraction between normal variables:When two normal variables are subtracted, the mean is the subtraction of the means while the standard deviation is the square root of the sum of the variances.

A sample of 243 brakes using Compound 1 yields an average brake life of 37,866 miles. The population standard deviation for Compound 1 is 4414 miles.This means that

A sample of 268 brakes using Compound 2 yields an average brake life of 45,789 miles. The population standard deviation for Compound 2 is 2368 miles.This means that

Distribution of the difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2.The mean is:

The standard deviation is:

Confidence interval:We have that to

findour level, that is the subtraction of 1 by the confidence interval divided by 2. So:Now, we have to find z in the Ztable as such z has a pvalue of .That is z with a pvalue of , so Z = 1.96.

Now, find the margin of error M as suchThe margin of error is of 623.2016.

The

lower endof the interval is the sample mean subtracted by M. So it is -7923 – 623.2016 = -8546.2016The

upper endof the interval is the sample mean added to M. So it is -7923 + 623.2016 = -7299.7984The 95% confidence interval for the true difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2 is (-8546.2016, -7299.7984).