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Sagot :
Let's address each part of the problem systematically.
### Part (a): Calculate and interpret a 95% confidence interval for the true average CO2 level
Given:
- Sample size (\( n \)) = 45
- Sample mean (\( \bar{x} \)) = 654.16 ppm
- Sample standard deviation (\( s \)) = 166.08 ppm
- Confidence level = 95%
Step 1: Determine the critical value (z-value) for a 95% confidence interval
For a 95% confidence level, the corresponding z-value is approximately 1.96 (using standard z-tables or statistical functions).
Step 2: Calculate the margin of error
The formula for the margin of error (E) in the context of a confidence interval for the mean is:
[tex]\[ E = z \times \left( \frac{s}{\sqrt{n}} \right) \][/tex]
Plugging in the given values:
[tex]\[ E = 1.96 \times \left( \frac{166.08}{\sqrt{45}} \right) \][/tex]
Step 3: Calculate the confidence interval
The confidence interval is given by:
[tex]\[ \bar{x} \pm E \][/tex]
So the lower bound and upper bound are:
[tex]\[ \text{Lower bound} = 654.16 - E \approx 605.64 \][/tex]
[tex]\[ \text{Upper bound} = 654.16 + E \approx 702.68 \][/tex]
Thus, the 95% confidence interval for the true average CO2 level is approximately (605.64 ppm, 702.68 ppm).
Interpretation:
We are 95% confident that this interval contains the true population mean.
### Part (b): Determine the necessary sample size for a desired interval width
Given:
- Desired interval width = 51 ppm
- Guessed standard deviation (\( \sigma \)) = 165 ppm
- Confidence level = 95%
Step 1: Determine the critical value (z-value) for a 95% confidence interval
From part (a), the z-value for a 95% confidence level is 1.96.
Step 2: Use the margin of error formula to find the required sample size
The margin of error \( E \) is half of the desired interval width:
[tex]\[ E = \frac{51}{2} = 25.5 \][/tex]
The formula to solve for \( n \) is derived from the margin of error formula:
[tex]\[ E = z \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
Rearranging to solve for \( n \):
[tex]\[ n = \left( \frac{z \times \sigma}{E} \right)^2 \][/tex]
Plugging in the given values:
[tex]\[ n = \left( \frac{1.96 \times 165}{25.5} \right)^2 \approx 160.46 \][/tex]
Since sample size must be an integer, we round up to the nearest whole number:
[tex]\[ n \approx 161 \][/tex]
Thus, the necessary sample size to achieve a desired interval width of 51 ppm at a 95% confidence level is 161 kitchens.
In conclusion:
- The 95% confidence interval for the true average CO2 level is approximately (605.64 ppm, 702.68 ppm). We are 95% confident that this interval contains the true population mean.
- To obtain a 95% confidence interval with a width of 51 ppm, the required sample size is 161 kitchens.
### Part (a): Calculate and interpret a 95% confidence interval for the true average CO2 level
Given:
- Sample size (\( n \)) = 45
- Sample mean (\( \bar{x} \)) = 654.16 ppm
- Sample standard deviation (\( s \)) = 166.08 ppm
- Confidence level = 95%
Step 1: Determine the critical value (z-value) for a 95% confidence interval
For a 95% confidence level, the corresponding z-value is approximately 1.96 (using standard z-tables or statistical functions).
Step 2: Calculate the margin of error
The formula for the margin of error (E) in the context of a confidence interval for the mean is:
[tex]\[ E = z \times \left( \frac{s}{\sqrt{n}} \right) \][/tex]
Plugging in the given values:
[tex]\[ E = 1.96 \times \left( \frac{166.08}{\sqrt{45}} \right) \][/tex]
Step 3: Calculate the confidence interval
The confidence interval is given by:
[tex]\[ \bar{x} \pm E \][/tex]
So the lower bound and upper bound are:
[tex]\[ \text{Lower bound} = 654.16 - E \approx 605.64 \][/tex]
[tex]\[ \text{Upper bound} = 654.16 + E \approx 702.68 \][/tex]
Thus, the 95% confidence interval for the true average CO2 level is approximately (605.64 ppm, 702.68 ppm).
Interpretation:
We are 95% confident that this interval contains the true population mean.
### Part (b): Determine the necessary sample size for a desired interval width
Given:
- Desired interval width = 51 ppm
- Guessed standard deviation (\( \sigma \)) = 165 ppm
- Confidence level = 95%
Step 1: Determine the critical value (z-value) for a 95% confidence interval
From part (a), the z-value for a 95% confidence level is 1.96.
Step 2: Use the margin of error formula to find the required sample size
The margin of error \( E \) is half of the desired interval width:
[tex]\[ E = \frac{51}{2} = 25.5 \][/tex]
The formula to solve for \( n \) is derived from the margin of error formula:
[tex]\[ E = z \times \left( \frac{\sigma}{\sqrt{n}} \right) \][/tex]
Rearranging to solve for \( n \):
[tex]\[ n = \left( \frac{z \times \sigma}{E} \right)^2 \][/tex]
Plugging in the given values:
[tex]\[ n = \left( \frac{1.96 \times 165}{25.5} \right)^2 \approx 160.46 \][/tex]
Since sample size must be an integer, we round up to the nearest whole number:
[tex]\[ n \approx 161 \][/tex]
Thus, the necessary sample size to achieve a desired interval width of 51 ppm at a 95% confidence level is 161 kitchens.
In conclusion:
- The 95% confidence interval for the true average CO2 level is approximately (605.64 ppm, 702.68 ppm). We are 95% confident that this interval contains the true population mean.
- To obtain a 95% confidence interval with a width of 51 ppm, the required sample size is 161 kitchens.
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