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Sagot :
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 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.
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 [tex]\mu_1 = 37866, s_1 = \frac{4414}{\sqrt{243}} = 283.16[/tex]
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 [tex]\mu_2 = 45789, s_2 = \frac{2368}{\sqrt{268}} = 144.65[/tex]
Distribution of the difference between average lifetimes for brakes using Compound 1 and brakes using Compound 2.
The mean is:
[tex]\mu = \mu_1 - \mu_2 = 37866 - 45789 = -7923[/tex]
The standard deviation is:
[tex]s = \sqrt{s_1^2 + s_2^2} = \sqrt{283.16^2 + 144.64^2} = 317.96[/tex]
Confidence interval:
We have that to find our [tex]\alpha[/tex] level, that is the subtraction of 1 by the confidence interval divided by 2. So:
[tex]\alpha = \frac{1 - 0.95}{2} = 0.025[/tex]
Now, we have to find z in the Ztable as such z has a pvalue of [tex]1 - \alpha[/tex].
That is z with a pvalue of [tex]1 - 0.025 = 0.975[/tex], so Z = 1.96.
Now, find the margin of error M as such
[tex]M = zs[/tex]
[tex]M = 1.96*317.96 = 623.2016[/tex]
The margin of error is of 623.2016.
The lower end of the interval is the sample mean subtracted by M. So it is -7923 - 623.2016 = -8546.2016
The upper end of the interval is the sample mean added to M. So it is -7923 + 623.2016 = -7299.7984
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).
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