Question

Past experience has indicated that the breaking strength of yarn used in manufacturing drapery material is normally distributed and that σ = 4 psi. A random sample of 11 specimens is tested, and the average breaking strength is found to be 97 psi. Find a 95% two-sided confidence interval on the true mean breaking strength. Round the answers to 1 decimal place.

Answer 1

95% two-sided confidence interval on the true mean breaking strength is (94.8cm, 99.2cm)

Explanation:

Our sample size is 11.

The first step to solve this problem is finding our degrees of freedom, that is, the sample size subtracted by 1. So

`df = 11-1 = 10`.

Then, we need to subtract one by the confidence level
`\alpha` and divide by 2. So:

`(1-0.95)/(2) = (0.05)/(2) = 0.025`

Now, we need our answers from both steps above to find a value T in the t-distribution table. So, with 10 and 0.025 in the two-sided t-distribution table, we have
`T = 1.812`

Now, we find the standard deviation of the sample. This is the division of the standard deviation by the square root of the sample size. So

`s = (4)/(√(11)) = 1.2060`

Now, we multiply T and s

cm

For the upper end of the interval, we add the sample mean and M. So the upper end of the interval here is

`L = 97 + 2.19 = 99.19 = 99.2`cm

So

95% two-sided confidence interval on the true mean breaking strength is (94.8cm, 99.2cm).