An investigator compares the durability of two different compounds used in the manufacture of a certain automobile brake lining. A sample of 127 brakes using Compound 1 yields an average brake life of 42,814 miles. A sample of 163 brakes using Compound 2 yields an average brake life of 37,197 miles. Assume that the population standard deviation for Compound 1 is 1819 miles, while the population standard deviation for Compound 2 is 1401 miles. Determine the 98% confidence interval for the true difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2. Step 1 of 3 is Point estimate so 42,814 - 37,197 = 5,617
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.
Step 3 of 3:
Construct the 98% confidence interval. Round your answers to the nearest whole number. (lower and upper endpoint)

The point estimate is 5,617.

The margin of error of a confidence interval for the difference between the two population means is 454.18386 .

The 98% confidence interval for the difference between the two population means is (5163, 6071).

Step-by-step explanation:

Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem establishes 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 difference of the means, while the standard deviation is the square root of the sum of the variances.

Compound 1:

127 brakes, average brake life of 42,814 miles, population standard deviation of 1819 miles. This means that:

Compound 2:

163 brakes, average brake life of 37,197 miles, population standard deviation of 1401 miles. This means that:

Distribution of the difference:

The point estimate is 5,617.

Confidence interval

The confidence interval is:

In which

z is the z-score that has a p-value of .

The margin of error is:

98% confidence level

So , z is the value of Z that has a p-value of , so .  

Margin of error:

The margin of error of a confidence interval for the difference between the two population means is 454.18386 .

For the confidence interval, as we round to the nearest whole number, we round it 454. So

The lower bound of the interval is:

The upper bound of the interval is:

The 98% confidence interval for the difference between the two population means is (5163, 6071).

8.75

how-to:

first, we need to find what we have to work with.

since i know that we have 7 (years) and 0.8 (inches), we can use these to solve and find the equation!

secondly, what you want to do, is see that we need to know how much growth out of the 0.8 inches he got in only 1 year, instead of 7 years.

let's try to divide.

7 ÷ 0.8 = 8.75