Answer:
The 90% confidence interval for the difference in mean breaking strength between the two types of yarn is (0.08N, 0.52N).
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 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
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
Subtraction of normal variables:
When we subtract two normal variables, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.
A sample of 125 pieces of yarn had mean breaking strength 6.1 N and standard deviation 0.7 N.
This means that
In a sample of 75 pieces of yarn from the new batch, the mean breaking strength was 5.8 N and the standard deviation was 1.0 N.
This means that
Distribution of the difference:
Confidence interval:
We have that to find our 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 = 1.645.
Now, find the margin of error M as such
The lower end of the interval is the sample mean subtracted by M. So it is 0.3 - 0.22 = 0.08N
The upper end of the interval is the sample mean added to M. So it is 0.3 + 0.22 = 0.52N
The 90% confidence interval for the difference in mean breaking strength between the two types of yarn is (0.08N, 0.52N).
