Question

ASK YOUR TEACHER An article reported that for a sample of 42 kitchens with gas cooking appliances monitored during a one-week period, the sample mean CO2 level (ppm) was 654.16, and the sample standard deviation was 165.23. (a) Calculate and interpret a 95% (two-sided) confidence interval for true average CO2 level in the population of all homes from which the sample was selected. (Round your answers to two decimal places.)

Answer 1

`654.16-2.02(165.23)/(√(42))=602.66`

`654.16+2.02(165.23)/(√(42))=705.66`

So on this case the 95% confidence interval would be given by (602.66;705.66)

Explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

`\bar X=654.16` represent the sample mean

`\mu` population mean (variable of interest)

s=165.23 represent the sample standard deviation

n=42 represent the sample size

Solution to the problem

The confidence interval for the mean is given by the following formula:

`\bar X \pm t_(\alpha/2)(s)/(√(n))` (1)

In order to calculate the critical value
`t_(\alpha/2)` we need to find first the degrees of freedom, given by:

`df=n-1=42-1=41`

Since the Confidence is 0.95 or 95%, the value of
`\alpha=0.05` and
`\alpha/2 =0.025`, and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,41)".And we see that
`t_(\alpha/2)=2.02`

Now we have everything in order to replace into formula (1):

`654.16-2.02(165.23)/(√(42))=602.66`

`654.16+2.02(165.23)/(√(42))=705.66`

So on this case the 95% confidence interval would be given by (602.66;705.66)