Thank you for visiting The body temperatures in F of 9 randomly selected men are recorded as follows 96 9 97 4 97 5 97 8 97 8 97. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
To construct a confidence interval for the mean body temperature of men based on a sample, we need to follow a few steps and use statistical formulas.
Step 1: Calculate the Sample Mean and Standard Deviation
The sample data consists of the following body temperatures in Fahrenheit ([tex]^{\circ}F[/tex]): 96.9, 97.4, 97.5, 97.8, 97.8, 97.9, 98.0, 98.1, and 98.6.
Sample Mean ([tex]\bar{x}[/tex]):
[tex]\bar{x} = \frac{\sum x_i}{n} = \frac{96.9 + 97.4 + 97.5 + 97.8 + 97.8 + 97.9 + 98.0 + 98.1 + 98.6}{9}[/tex]
After calculating, [tex]\bar{x} \approx 97.8^{\circ}F[/tex].Sample Standard Deviation ([tex]s[/tex]):
To find the standard deviation, calculate:
[tex]s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}[/tex]
After calculations, [tex]s \approx 0.52[/tex].
Step 2: Find the Critical t-value for 95% Confidence Level
Since the sample size [tex]n = 9[/tex], degrees of freedom [tex]df = n - 1 = 8[/tex].
Using a t-distribution table or calculator, find the critical t-value for a 95% confidence level (one-tailed), which is approximately [tex]t_{0.05, 8} \approx 1.860[/tex].
Step 3: Calculate the Margin of Error (ME)
[tex]ME = t \cdot \frac{s}{\sqrt{n}} = 1.860 \cdot \frac{0.52}{\sqrt{9}} = 1.860 \cdot 0.1733 \approx 0.32[/tex]
Step 4: Construct the Confidence Interval
Since the question asks for testing if the mean is greater than 97.5°F, we'll construct a one-tailed confidence interval:
[tex]\text{Lower Limit} = \bar{x} + ME = 97.8 + 0.32 = 98.12[/tex]
Thus, the one-tailed 95% confidence interval is all values greater than 98.12°F.
Conclusion
The calculated confidence interval does not include the value 97.5°F. Therefore, we have enough evidence at the 0.05 significance level to support the claim that the mean body temperature of men is greater than 97.5°F.
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