Find the information you're looking for at Westonci.ca, the trusted Q&A platform with a community of knowledgeable experts. Discover detailed solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To construct a 99% confidence interval for the mean body temperature of all healthy humans, we need to follow these steps:
1. Identify the given data:
- Sample size (\(n\)) = 110
- Sample mean (\(\bar{x}\)) = 98.0°F
- Sample standard deviation (\(s\)) = 0.74°F
- Confidence level = 99%
2. Determine the degrees of freedom (df):
- Degrees of freedom (\(df\)) = \(n - 1\) = 110 - 1 = 109
3. Find the t-critical value:
- For a 99% confidence level and 109 degrees of freedom. Based on tabulated values or computational tools, we find the t-critical value (\(t_{\alpha/2}\)) to be approximately 2.622.
4. Calculate the margin of error (ME):
- The formula for the margin of error is:
[tex]\[ \text{ME} = t_{\alpha/2} \times \left(\frac{s}{\sqrt{n}}\right) \][/tex]
- Plug in the values:
[tex]\[ \text{ME} = 2.622 \times \left(\frac{0.74}{\sqrt{110}}\right) \][/tex]
- Calculate the standard error (SE):
[tex]\[ \text{SE} = \frac{0.74}{\sqrt{110}} \approx 0.0704 \][/tex]
- Now, calculate the margin of error:
[tex]\[ \text{ME} \approx 2.622 \times 0.0704 \approx 0.185 \][/tex]
5. Construct the confidence interval:
- The lower limit of the confidence interval is:
[tex]\[ \text{Lower limit} = \bar{x} - \text{ME} = 98.0 - 0.185 \approx 97.815 \][/tex]
- The upper limit of the confidence interval is:
[tex]\[ \text{Upper limit} = \bar{x} + \text{ME} = 98.0 + 0.185 \approx 98.185 \][/tex]
6. State the confidence interval:
- The 99% confidence interval for the mean body temperature of all healthy humans is approximately:
[tex]\[ 97.815^\circ F < \mu < 98.185^\circ F \][/tex]
Interpretation about the use of \(98.6^\circ F\):
Considering the constructed confidence interval of [tex]\(97.815^\circ F < \mu < 98.185^\circ F\)[/tex], we observe that 98.6°F is not within this interval. This suggests that the traditional average body temperature of 98.6°F might not be an accurate estimate of the mean body temperature for healthy humans based on the provided sample.
1. Identify the given data:
- Sample size (\(n\)) = 110
- Sample mean (\(\bar{x}\)) = 98.0°F
- Sample standard deviation (\(s\)) = 0.74°F
- Confidence level = 99%
2. Determine the degrees of freedom (df):
- Degrees of freedom (\(df\)) = \(n - 1\) = 110 - 1 = 109
3. Find the t-critical value:
- For a 99% confidence level and 109 degrees of freedom. Based on tabulated values or computational tools, we find the t-critical value (\(t_{\alpha/2}\)) to be approximately 2.622.
4. Calculate the margin of error (ME):
- The formula for the margin of error is:
[tex]\[ \text{ME} = t_{\alpha/2} \times \left(\frac{s}{\sqrt{n}}\right) \][/tex]
- Plug in the values:
[tex]\[ \text{ME} = 2.622 \times \left(\frac{0.74}{\sqrt{110}}\right) \][/tex]
- Calculate the standard error (SE):
[tex]\[ \text{SE} = \frac{0.74}{\sqrt{110}} \approx 0.0704 \][/tex]
- Now, calculate the margin of error:
[tex]\[ \text{ME} \approx 2.622 \times 0.0704 \approx 0.185 \][/tex]
5. Construct the confidence interval:
- The lower limit of the confidence interval is:
[tex]\[ \text{Lower limit} = \bar{x} - \text{ME} = 98.0 - 0.185 \approx 97.815 \][/tex]
- The upper limit of the confidence interval is:
[tex]\[ \text{Upper limit} = \bar{x} + \text{ME} = 98.0 + 0.185 \approx 98.185 \][/tex]
6. State the confidence interval:
- The 99% confidence interval for the mean body temperature of all healthy humans is approximately:
[tex]\[ 97.815^\circ F < \mu < 98.185^\circ F \][/tex]
Interpretation about the use of \(98.6^\circ F\):
Considering the constructed confidence interval of [tex]\(97.815^\circ F < \mu < 98.185^\circ F\)[/tex], we observe that 98.6°F is not within this interval. This suggests that the traditional average body temperature of 98.6°F might not be an accurate estimate of the mean body temperature for healthy humans based on the provided sample.
Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.