Question

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)

Answer 1

Answer:

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 $\mu$ 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.  

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:

$\mu_1 = 42814$

$s_1 = \frac{1819}{\sqrt{127}} = 161.41$

Compound 2:

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

$\mu_2 = 37197$

$s_2 = \frac{1401}{\sqrt{163}} = 109.73$

Distribution of the difference:

$\mu = \mu_1 - \mu_2 = 42814 - 37197 = 5617$

The point estimate is 5,617.

$s = \sqrt{s_1^2 + s_2^2} = \sqrt{161.41^2 + 109.73^2} = 195.18$

Confidence interval

The confidence interval is:

$\mu \pm zs$

In which

z is the z-score that has a p-value of $1 - \frac{\alpha}{2}$.

The margin of error is:

$M = zs$

98% confidence level

So $\alpha = 0.02$, z is the value of Z that has a p-value of $1 - \frac{0.02}{2} = 0.99$, so $Z = 2.327$.  

Margin of error:

$M = zs = 195.18*2.327 = 454.18386$

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:

$\mu - zs = \mu - M = 5617 - 454 = 5163$

The upper bound of the interval is:

$\mu + zs = \mu + M = 5617 + 454 = 6071$

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