Question

The pH levels of a random sample of 16 chemical mixtures from a process were measured, and a sample mean ¯x = 6.861 and a sample standard deviation s = 0.440 were obtained. The scientists presented a confidence interval (6.668, [infinity]) for the average pH level of chemical mixtures from the process. What is the confidence level of this confidence interval? Suppose the pH level of the chemical mixture is normally distributed.

Answer 1

(6.668, [infinity])

So then we have this condition:

`6.668 = 6.861 -t_(\alpha/2) (0.440)/(√(16))`

And if we solve for the critical value we got:

`t_(\alpha/2)= (√(16) *(6.861-6.668))/(0.440) = 1.755`

And we can find the probability accumulated in the left of 1.755 with a distribution with degrees of freedom
`df = n-1= 16-1 = 15` and we got: 0.95, with the following excel code for example:

"=T.DIST(1.755,15,TRUE)"

So then the confidence level would be 95%

Explanation:

For this case we have the following info:

`\bar X = 6.861` represent the sample mean

`s = 0.440` represent the sample deviation

n =16 represent the sample size

For this case the scientists calculate a lower bound confidence interval given by:

`(\bar X -t_(\alpha/2) (s)/(√(n)) , \infty)`

And the interval given is:

(6.668, [infinity])

So then we have this condition:

`6.668 = 6.861 -t_(\alpha/2) (0.440)/(√(16))`

And if we solve for the critical value we got:

`t_(\alpha/2)= (√(16) *(6.861-6.668))/(0.440) = 1.755`

And we can find the probability accumulated in the left of 1.755 with a distribution with degrees of freedom
`df = n-1= 16-1 = 15` and we got: 0.95, with the following excel code for example:

"=T.DIST(1.755,15,TRUE)"

So then the confidence level would be 95%