Questions: A data set includes 106 body temperatures of healthy adult humans having a mean of 98.7°F and a standard deviation of 0.66°F. Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6°F as the mean body temperature? What is the confidence interval estimate of the population mean μ ? °F<μ< °F (Round to three decimal places as needed.)

A data set includes 106 body temperatures of healthy adult humans having a mean of 98.7°F and a standard deviation of 0.66°F. Construct a 99% confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of 98.6°F as the mean body temperature?

What is the confidence interval estimate of the population mean μ ?
°F<μ< °F
(Round to three decimal places as needed.)

Transcript text: A data set includes 106 body temperatures of healthy adult humans having a mean of $98.7^{\circ} \mathrm{F}$ and a standard deviation of $0.66^{\circ} \mathrm{F}$. Construct a $99 \%$ confidence interval estimate of the mean body temperature of all healthy humans. What does the sample suggest about the use of $98.6^{\circ} \mathrm{F}$ as the mean body temperature? What is the confidence interval estimate of the population mean $\mu$ ? $\square$ $7^{\circ} \mathrm{F}<\mu<$ $\square$ ${ }^{\circ} \mathrm{F}$ (Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Given Data

We have a sample of body temperatures of healthy adult humans with the following statistics:

Sample mean ($\bar{x}$): $98.7^{\circ} \mathrm{F}$
Sample size ($n$): $106$
Sample standard deviation ($s$): $0.66^{\circ} \mathrm{F}$
Confidence level: $99\%$

Step 2: Calculate the Z-Score

For a $99\%$ confidence level, the Z-score corresponding to the critical value is: \[ z = 2.576 \]

Step 3: Calculate the Margin of Error

The margin of error ($E$) is calculated using the formula: \[ E = z \cdot \frac{s}{\sqrt{n}} = 2.576 \cdot \frac{0.66}{\sqrt{106}} \approx 0.165 \]

Step 4: Construct the Confidence Interval

The confidence interval for the population mean ($\mu$) is given by: \[ \bar{x} \pm E \] Calculating the lower and upper bounds: \[ \text{Lower bound} = 98.7 - 0.165 \approx 98.535 \] \[ \text{Upper bound} = 98.7 + 0.165 \approx 98.865 \]

Thus, the $99\%$ confidence interval is: \[ (98.535, 98.865) \]

Step 5: Interpretation of the Confidence Interval

The confidence interval suggests that the true mean body temperature of all healthy humans is likely to fall within the range: \[ 98.535^{\circ} \mathrm{F} < \mu < 98.865^{\circ} \mathrm{F} \]

Since $98.6^{\circ} \mathrm{F}$ lies within this interval, we conclude that the sample suggests that $98.6^{\circ} \mathrm{F}$ is a plausible mean body temperature for healthy humans.