To construct a [tex]\(99\%\)[/tex] confidence interval estimate of the mean body temperature of all healthy humans, we'll follow these steps:
1. Identify the sample statistics and the confidence level:
- Sample size ([tex]\(n\)[/tex]): 108
- Sample mean ([tex]\(\bar{x}\)[/tex]): [tex]\(98.2^{\circ} F\)[/tex]
- Sample standard deviation ([tex]\(s\)[/tex]): [tex]\(0.64^{\circ} F\)[/tex]
- Confidence level: [tex]\(99\%\)[/tex]
2. Determine the critical value for the [tex]\(99\%\)[/tex] confidence level:
- The confidence level [tex]\(99\%\)[/tex] means that [tex]\(\alpha = 1 - 0.99 = 0.01\)[/tex].
- Since the sample size [tex]\(n = 108\)[/tex] is large, we use the t-distribution with [tex]\(df = n - 1 = 107\)[/tex].
- The critical value [tex]\(t_{\alpha/2}\)[/tex] for [tex]\(99\%\)[/tex] confidence level from the t-distribution with 107 degrees of freedom is approximately [tex]\(2.623\)[/tex].
3. Calculate the margin of error (ME):
[tex]\[
\text{ME} = t_{\alpha/2} \times \frac{s}{\sqrt{n}}
\][/tex]
Plug in the values:
[tex]\[
\text{ME} = 2.623 \times \frac{0.64}{\sqrt{108}} \approx 2.623 \times 0.0615 \approx 0.162
\][/tex]
4. Compute the confidence interval:
[tex]\[
\text{Lower limit} = \bar{x} - \text{ME} = 98.2 - 0.162 = 98.038
\][/tex]
[tex]\[
\text{Upper limit} = \bar{x} + \text{ME} = 98.2 + 0.162 = 98.362
\][/tex]
Therefore, the [tex]\(99\%\)[/tex] confidence interval estimate of the population mean [tex]\(\mu\)[/tex] is:
[tex]\[
98.038^{\circ} F < \mu < 98.362^{\circ} F
\][/tex]
Now let's interpret what this confidence interval suggests about the use of [tex]\(98.6^{\circ} F\)[/tex] as the mean body temperature. Since [tex]\(98.6^{\circ} F\)[/tex] is outside the confidence interval [tex]\(98.038^{\circ} F < \mu < 98.362^{\circ} F\)[/tex], the sample suggests that [tex]\(98.6^{\circ} F\)[/tex] is not a plausible value for the mean body temperature of all healthy humans.
