Question

A sample of 125 pieces of yarn had mean breaking strength 6.1 N and standard deviation 0.7 N. A new batch of yarn was made, using new raw materials from a different vendor. 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. Find a 90% confidence interval for the difference in mean breaking strength between the two types of yarn.

Answer 1

The 90% confidence interval for the difference in mean breaking strength between the two types of yarn is (0.08N, 0.52N).

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
`\mu` and standard deviation
`\sigma`, 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
`\mu_1 = 6.1, s_1 = (0.7)/(√(125)) = 0.0626`

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
`\mu_2 = 5.8, s_2 = (1)/(√(75)) = 0.1155`

Distribution of the difference:

`\mu = \mu_1 - \mu_2 = 6.1 - 5.8 = 0.3`

`s = √(s_1^2+s_2^2) = √(0.0626^2+0.1155^2) = 0.1314`

Confidence interval:

We have that to find our
`\alpha` level, that is the subtraction of 1 by the confidence interval divided by 2. So:

`\alpha = (1 - 0.9)/(2) = 0.05`

Now, we have to find z in the Ztable as such z has a pvalue of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.05 = 0.95`, so Z = 1.645.

Now, find the margin of error M as such

`M = zs`

`M = 1.645*0.1314 = 0.22`

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).