A data set includes 110 body temperatures of healthy adult humans, with a mean of 98.1°F and a standard deviation of 0.64°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? What should replace the question marks? Round to three decimal places. ?°F < μ < ?°F What does this suggest about the use of 98.6°F as the mean body temperature? Which letter is correct? A. This suggests that the mean body temperature could very possibly be 98.6°F. B. This suggests that the mean body temperature could be higher than 98.6°F. C. This suggests that the mean body temperature could be lower than 98.6°F.
The Correct Answer and Explanation is:
Given:
- Sample size, n = 110
- Sample mean, x̄ = 98.1°F
- Sample standard deviation, s = 0.64°F
- Confidence level = 99%
Step 1: Use the t-distribution
Since the population standard deviation is unknown and the sample size is large (>30), we can use the t-distribution.
Degrees of freedom:
df = n – 1 = 109
For a 99% confidence level and 109 degrees of freedom, the t-critical value (t*) ≈ 2.626 (from a t-table or calculator).
Step 2: Calculate the standard error (SE)
SE=sn=0.64110≈0.6410.4881≈0.061SE = \frac{s}{\sqrt{n}} = \frac{0.64}{\sqrt{110}} ≈ \frac{0.64}{10.4881} ≈ 0.061SE=ns=1100.64≈10.48810.64≈0.061
Step 3: Construct the confidence interval
CI=xˉ±t∗⋅SE=98.1±2.626⋅0.061≈98.1±0.160CI = x̄ \pm t^* \cdot SE = 98.1 \pm 2.626 \cdot 0.061 ≈ 98.1 \pm 0.160CI=xˉ±t∗⋅SE=98.1±2.626⋅0.061≈98.1±0.160CI≈(97.940,98.260)CI ≈ (97.940, 98.260)CI≈(97.940,98.260)
Final Answer:
97.940°F < μ < 98.260°F
Interpretation and Analysis
The calculated 99% confidence interval for the population mean body temperature is approximately (97.940°F, 98.260°F). This interval means we are 99% confident that the true average body temperature of all healthy adults falls within this range.
Historically, 98.6°F has been cited as the standard average human body temperature. However, this value does not fall within the computed confidence interval. Because the entire interval lies below 98.6°F, this suggests that the widely accepted average of 98.6°F may be an overestimate of the true mean.
The data set, derived from a sample of 110 healthy adults, provides strong evidence that the actual mean body temperature is lower than 98.6°F. Over time, multiple studies have challenged the 98.6°F standard, and this result further supports the notion that the average body temperature in modern populations is closer to 98.1°F or even slightly less.
Therefore, based on this confidence interval and statistical reasoning, we can infer that the mean body temperature for healthy adults is likely less than 98.6°F, and the historical standard may need to be updated in clinical practice.
Correct Choice:
C. This suggests that the mean body temperature could be lower than 98.6°F.
