(a) Calculating the 95% Confidence Interval for the True Mean CO₂ Level

To calculate a two-sided confidence interval for the population mean, we use the formula:

$\text{CI} = \bar{x} \pm t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$

Where:

• $\bar{x} = 654.16$ (sample mean)
• $s = 165.4$ (sample standard deviation)
• $n = 48$ (sample size)
• $t_{\alpha/2}$: critical value for 95% confidence level with $df = n - 1 = 47$.

Step 1: Find $t_{\alpha/2}$

Using a $t$-distribution table or software, the $t_{\alpha/2}$ value for $df = 47$ at 95% confidence is approximately $2.011$.

Step 2: Calculate the Margin of Error (ME)

$\text{ME} = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ $\text{ME} = 2.011 \cdot \frac{165.4}{\sqrt{48}}$ $\text{ME} = 2.011 \cdot 23.881 \approx 48.03$

Step 3: Confidence Interval

$\text{CI} = 654.16 \pm 48.03$ $\text{CI} = (606.13, 702.19)$

Interpretation: We are 95% confident that the true average CO₂ level in the population of homes is between 606.13 ppm and 702.19 ppm.

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(b) Determining Sample Size for Desired Interval Width

The formula for the margin of error in a confidence interval is: $\text{ME} = t_{\alpha/2} \cdot \frac{s}{\sqrt{n}}$ To achieve a desired interval width $W = 48$, the margin of error is half the width: $\text{ME} = \frac{W}{2} = \frac{48}{2} = 24$

Step 1: Rearrange the Formula for $n$

$n = \left(\frac{t_{\alpha/2} \cdot s}{\text{ME}}\right)^2$

Step 2: Plug in Values

Using $t_{\alpha/2} = 2.011$, $s = 165$, and $\text{ME} = 24$: $n = \left(\frac{2.011 \cdot 165}{24}\right)^2$ $n = \left(\frac{331.815}{24}\right)^2 \approx (13.826)^2 \approx 191.23$

Since sample size must be an integer, round up to the nearest whole number: $n = 192$

Answer: A sample size of 192 kitchens would be necessary to achieve a 95% confidence interval width of 48 ppm.

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Recap and Questions

• (a) 95% Confidence Interval: $606.13 \, \text{ppm} \, \text{to} \, 702.19 \, \text{ppm}$
• (b) Required Sample Size: $n = 192$

1. Why is the $t$-distribution used instead of the normal distribution here?
2. What factors influence the width of a confidence interval?
3. How does increasing the sample size affect the confidence interval?
4. What assumptions are made when calculating the confidence interval?
5. How would the interval change if the sample standard deviation were smaller?

Tip: Confidence intervals provide a range of plausible values for the population mean, but they rely on the assumption of a representative sample.