Westonci.ca is your trusted source for finding answers to all your questions. Ask, explore, and learn with our expert community. Explore a wealth of knowledge from professionals across different disciplines on our comprehensive platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
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.
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.