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Answer :
To find the 99% confidence interval for the mean body temperature of adults in the town, we follow these steps:
1. Collect Data: We have the following body temperature data in degrees Fahrenheit:
99.3, 97.8, 98.0, 97.1, 99.5, 99.0, 98.5, 98.7, 98.1, 99.7, 99.1, 99.8
2. Calculate the Sample Mean:
The sample mean is the average of all the data points. Add all the temperatures and divide by the number of data points.
[tex]\[
\text{Sample Mean} = \frac{99.3 + 97.8 + 98.0 + 97.1 + 99.5 + 99.0 + 98.5 + 98.7 + 98.1 + 99.7 + 99.1 + 99.8}{12} = 98.717
\][/tex]
3. Calculate the Sample Standard Deviation:
Use the formula for the sample standard deviation. This involves finding the difference of each temperature from the mean, squaring it, summing these squares, dividing by the number of data points minus one, and then taking the square root of that result.
[tex]\[
\text{Sample Standard Deviation} \approx 0.844
\][/tex]
4. Determine the Sample Size (n):
In this case, the sample size is [tex]\( n = 12 \)[/tex] because there are 12 temperature readings.
5. Determine the Z-Score for 99% Confidence Level:
For a 99% confidence level, we use a Z-score that corresponds to the central 99% of a standard normal distribution. The Z-score for 99% confidence is approximately 2.576.
6. Calculate the Margin of Error:
The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = Z \times \left(\frac{\text{Sample Standard Deviation}}{\sqrt{n}}\right)
\][/tex]
[tex]\[
\text{Margin of Error} = 2.576 \times \left(\frac{0.844}{\sqrt{12}}\right) \approx 0.622
\][/tex]
7. Calculate the Confidence Interval:
Now, we find the lower and upper bounds of the confidence interval using the sample mean and the margin of error:
[tex]\[
\text{Lower Bound} = \text{Sample Mean} - \text{Margin of Error} = 98.717 - 0.622 = 98.095
\][/tex]
[tex]\[
\text{Upper Bound} = \text{Sample Mean} + \text{Margin of Error} = 98.717 + 0.622 = 99.339
\][/tex]
Thus, the 99% confidence interval for the mean body temperature is [tex]\((98.095, 99.339)\)[/tex].
1. Collect Data: We have the following body temperature data in degrees Fahrenheit:
99.3, 97.8, 98.0, 97.1, 99.5, 99.0, 98.5, 98.7, 98.1, 99.7, 99.1, 99.8
2. Calculate the Sample Mean:
The sample mean is the average of all the data points. Add all the temperatures and divide by the number of data points.
[tex]\[
\text{Sample Mean} = \frac{99.3 + 97.8 + 98.0 + 97.1 + 99.5 + 99.0 + 98.5 + 98.7 + 98.1 + 99.7 + 99.1 + 99.8}{12} = 98.717
\][/tex]
3. Calculate the Sample Standard Deviation:
Use the formula for the sample standard deviation. This involves finding the difference of each temperature from the mean, squaring it, summing these squares, dividing by the number of data points minus one, and then taking the square root of that result.
[tex]\[
\text{Sample Standard Deviation} \approx 0.844
\][/tex]
4. Determine the Sample Size (n):
In this case, the sample size is [tex]\( n = 12 \)[/tex] because there are 12 temperature readings.
5. Determine the Z-Score for 99% Confidence Level:
For a 99% confidence level, we use a Z-score that corresponds to the central 99% of a standard normal distribution. The Z-score for 99% confidence is approximately 2.576.
6. Calculate the Margin of Error:
The margin of error is calculated using the formula:
[tex]\[
\text{Margin of Error} = Z \times \left(\frac{\text{Sample Standard Deviation}}{\sqrt{n}}\right)
\][/tex]
[tex]\[
\text{Margin of Error} = 2.576 \times \left(\frac{0.844}{\sqrt{12}}\right) \approx 0.622
\][/tex]
7. Calculate the Confidence Interval:
Now, we find the lower and upper bounds of the confidence interval using the sample mean and the margin of error:
[tex]\[
\text{Lower Bound} = \text{Sample Mean} - \text{Margin of Error} = 98.717 - 0.622 = 98.095
\][/tex]
[tex]\[
\text{Upper Bound} = \text{Sample Mean} + \text{Margin of Error} = 98.717 + 0.622 = 99.339
\][/tex]
Thus, the 99% confidence interval for the mean body temperature is [tex]\((98.095, 99.339)\)[/tex].
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