What is the Value of Y?
2 answers:
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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 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).