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Answer :
Sure! Let's walk through the steps to construct a 98% confidence interval estimate for the true mean body temperature of the population based on the given data.
Step 1: Point Estimate (Sample Mean)
To find the sample mean, sum up all the given body temperatures and divide by the number of temperatures.
1. Add up the temperatures: [tex]\(98.6 + 98.6 + 98.4 + 97.2 + 99.1 + 98.4 + 97.9 + 99.5 = 787.7\)[/tex]
2. Divide by the number of temperatures (n = 8): [tex]\(\bar{x} = \frac{787.7}{8} = 98.462\)[/tex]
Point estimate [tex]\(\bar{x}= 98.462\)[/tex] (rounded to 3 decimal places)
Step 2: Critical Value ([tex]\(t_{\frac{α}{2}}\)[/tex])
To find the critical value for a 98% confidence interval with a sample size of 8, identify the t-score with 7 degrees of freedom (n-1) for a 98% confidence level.
Critical value [tex]\(t_{\frac{α}{2}} = 2.998\)[/tex] (rounded to 3 decimal places)
Step 3: Standard Error
The standard error measures the dispersion of sample means around the population mean.
1. Compute the sample standard deviation (s).
2. The formula for standard error is [tex]\(\frac{s}{\sqrt{n}}\)[/tex].
For this data:
Standard error = 0.248 (rounded to 3 decimal places)
Step 4: Margin of Error
The margin of error combines the critical value and the standard error to provide the range of the confidence interval.
1. Use the formula: Margin of error = [tex]\(t_{\frac{α}{2}} \times \frac{s}{\sqrt{n}}\)[/tex].
With the numbers:
Margin of error = 0.743 (rounded to 3 decimal places)
Step 5: Confidence Interval
Now, we calculate the confidence interval using the point estimate and the margin of error.
1. Subtract the margin of error from the point estimate to get the lower bound:
[tex]\(98.462 - 0.743 = 97.719\)[/tex]
2. Add the margin of error to the point estimate to get the upper bound:
[tex]\(98.462 + 0.743 = 99.205\)[/tex]
Finally, express the confidence interval with 1 decimal place of accuracy as:
Confidence Interval = (97.7, 99.2)
This interval suggests that we are 98% confident that the true mean body temperature of the population is between 97.7°F and 99.2°F.
Step 1: Point Estimate (Sample Mean)
To find the sample mean, sum up all the given body temperatures and divide by the number of temperatures.
1. Add up the temperatures: [tex]\(98.6 + 98.6 + 98.4 + 97.2 + 99.1 + 98.4 + 97.9 + 99.5 = 787.7\)[/tex]
2. Divide by the number of temperatures (n = 8): [tex]\(\bar{x} = \frac{787.7}{8} = 98.462\)[/tex]
Point estimate [tex]\(\bar{x}= 98.462\)[/tex] (rounded to 3 decimal places)
Step 2: Critical Value ([tex]\(t_{\frac{α}{2}}\)[/tex])
To find the critical value for a 98% confidence interval with a sample size of 8, identify the t-score with 7 degrees of freedom (n-1) for a 98% confidence level.
Critical value [tex]\(t_{\frac{α}{2}} = 2.998\)[/tex] (rounded to 3 decimal places)
Step 3: Standard Error
The standard error measures the dispersion of sample means around the population mean.
1. Compute the sample standard deviation (s).
2. The formula for standard error is [tex]\(\frac{s}{\sqrt{n}}\)[/tex].
For this data:
Standard error = 0.248 (rounded to 3 decimal places)
Step 4: Margin of Error
The margin of error combines the critical value and the standard error to provide the range of the confidence interval.
1. Use the formula: Margin of error = [tex]\(t_{\frac{α}{2}} \times \frac{s}{\sqrt{n}}\)[/tex].
With the numbers:
Margin of error = 0.743 (rounded to 3 decimal places)
Step 5: Confidence Interval
Now, we calculate the confidence interval using the point estimate and the margin of error.
1. Subtract the margin of error from the point estimate to get the lower bound:
[tex]\(98.462 - 0.743 = 97.719\)[/tex]
2. Add the margin of error to the point estimate to get the upper bound:
[tex]\(98.462 + 0.743 = 99.205\)[/tex]
Finally, express the confidence interval with 1 decimal place of accuracy as:
Confidence Interval = (97.7, 99.2)
This interval suggests that we are 98% confident that the true mean body temperature of the population is between 97.7°F and 99.2°F.
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