Answer :
To find the 95% confidence interval for the mean body temperature of adults in the town based on the given data, follow these steps:
1. Collect the Data: The body temperatures provided are:
- [tex]\( 97.7, 99.3, 96.4, 99.9, 98.2, 98.1, 96.5, 98.8, 97.5, 96.8, 99.5, 98.4 \)[/tex]
2. Calculate the Sample Mean:
- Add all the temperatures together and divide by the number of data points (12 in this case).
- The sample mean is calculated as approximately [tex]\( 98.092 \)[/tex].
3. Calculate the Sample Standard Deviation:
- Use the formula for the sample standard deviation, which accounts for a degree of freedom (n-1 where n is the sample size).
- The sample standard deviation is approximately [tex]\( 1.163 \)[/tex].
4. Determine the Sample Size (n):
- There are 12 data points in the sample, so [tex]\( n = 12 \)[/tex].
5. Find the t-score for the 95% Confidence Level:
- With a confidence level of 95% and 11 degrees of freedom (n-1), find the t-score using a t-distribution table.
- The t-score is approximately [tex]\( 2.201 \)[/tex].
6. Calculate the Margin of Error:
- The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = t \times \left(\frac{\text{Sample Standard Deviation}}{\sqrt{n}}\right)
\][/tex]
- Substituting the values, the margin of error is approximately [tex]\( 0.739 \)[/tex].
7. Determine the Confidence Interval:
- Calculate the lower bound by subtracting the margin of error from the sample mean.
- Calculate the upper bound by adding the margin of error to the sample mean.
- The 95% confidence interval is approximately [tex]\((97.352, 98.831)\)[/tex].
Therefore, the 95% confidence interval for the mean body temperature of adults in this town is [tex]\((97.352, 98.831)\)[/tex] when rounded to three decimal places.
1. Collect the Data: The body temperatures provided are:
- [tex]\( 97.7, 99.3, 96.4, 99.9, 98.2, 98.1, 96.5, 98.8, 97.5, 96.8, 99.5, 98.4 \)[/tex]
2. Calculate the Sample Mean:
- Add all the temperatures together and divide by the number of data points (12 in this case).
- The sample mean is calculated as approximately [tex]\( 98.092 \)[/tex].
3. Calculate the Sample Standard Deviation:
- Use the formula for the sample standard deviation, which accounts for a degree of freedom (n-1 where n is the sample size).
- The sample standard deviation is approximately [tex]\( 1.163 \)[/tex].
4. Determine the Sample Size (n):
- There are 12 data points in the sample, so [tex]\( n = 12 \)[/tex].
5. Find the t-score for the 95% Confidence Level:
- With a confidence level of 95% and 11 degrees of freedom (n-1), find the t-score using a t-distribution table.
- The t-score is approximately [tex]\( 2.201 \)[/tex].
6. Calculate the Margin of Error:
- The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = t \times \left(\frac{\text{Sample Standard Deviation}}{\sqrt{n}}\right)
\][/tex]
- Substituting the values, the margin of error is approximately [tex]\( 0.739 \)[/tex].
7. Determine the Confidence Interval:
- Calculate the lower bound by subtracting the margin of error from the sample mean.
- Calculate the upper bound by adding the margin of error to the sample mean.
- The 95% confidence interval is approximately [tex]\((97.352, 98.831)\)[/tex].
Therefore, the 95% confidence interval for the mean body temperature of adults in this town is [tex]\((97.352, 98.831)\)[/tex] when rounded to three decimal places.
