Question

The concentration of carbon monoxide (CO) in a gas sample is measured by a spectrophotometer and found to be 85 ppm. Through long experience with this instrument, it is believed that its measurements are unbiased and normally distributed, with an uncertainty (standard deviation) of 9 ppm. Find a 95% confidence interval for the concentration of CO in this sample. Round the answers to two decimal places. The 95% confidence interval is

Answer 1

The confidence interval is
`(85 - (14.81)/(√(n)),85 + (14.81)/(√(n)))`, in which n is the size of the sample.

Explanation:

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

`\alpha = (1 - 0.95)/(2) = 0.025`

Now, we have to find z in the Z-table as such z has a p-value of
`1 - \alpha`.

That is z with a pvalue of
`1 - 0.025 = 0.975`, so Z = 1.96.

Now, find the margin of error M as such

`M = z(\sigma)/(√(n))`

In which
`\sigma` is the standard deviation of the population and n is the size of the sample.

`M = 1.645(9)/(√(n)) = (14.81)/(√(n))`

The lower end of the interval is the sample mean subtracted by M. So it is
`85 - (14.81)/(√(n))`

The upper end of the interval is the sample mean added to M. So it is
`85 + (14.81)/(√(n))`

The confidence interval is
`(85 - (14.81)/(√(n)),85 + (14.81)/(√(n)))`, in which n is the size of the sample.