Question

An article reported that for a sample of 52 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 165.4.

Required:
a. Calculate and interpret a 9596 (two-sided) confidence interval for true average C02 level in the population of all homes from which the sample was selected.
b. Suppose the investigators had made a rough guess of 175 for the value of s before collecting data. What sample size would be necessary to obtain an interval width of 50 ppm for a confidence level of 95%?

Answer 1

a)
`654.16-2.01(165.4)/(√(52))=608.06`

`654.16+2.01(165.4)/(√(52))=700.26`

b)
`n=((1.960(175))/(25))^2 =188.23 \approx 189`

So the answer for this case would be n=189 rounded up to the nearest integer

Explanation:

Part a

`\bar X=654.16` represent the sample mean

`\mu` population mean (variable of interest)

s=165.4 represent the sample standard deviation

n =52represent the sample size

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

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

The degrees of freedom aregiven by:

`df=n-1=52-1=51`

Since the Confidence is 0.95 or 95%, the significance
`\alpha=0.05` and
`\alpha/2 =0.025`, and the critical value would be
`t_(\alpha/2)=2.01`

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

`654.16-2.01(165.4)/(√(52))=608.06`

`654.16+2.01(165.4)/(√(52))=700.26`

Part b

The margin of error is given by this formula:

`ME=z_(\alpha/2)(s)/(√(n))` (a)

And on this case we have that ME =25 and we are interested in order to find the value of n, if we solve n from equation (a) we got:

`n=((z_(\alpha/2) s)/(ME))^2` (b)

The critical value for this case wuld be
`z_(\alpha/2)=1.960`, replacing into formula (b) we got:

`n=((1.960(175))/(25))^2 =188.23 \approx 189`

So the answer for this case would be n=189 rounded up to the nearest integer