Thank you for visiting Listed below are body temperatures from five different subjects measured at 8 AM and again at 12 AM Find the values of tex overline d. 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 solve the problem of finding [tex]\(\overline{d}\)[/tex] and [tex]\(s_d\)[/tex] for the listed body temperatures, follow these steps:
1. Identify the Paired Differences:
- We have temperatures for five subjects at 8 AM and 12 AM.
- Calculate the difference for each subject by subtracting the 8 AM temperature from the 12 AM temperature.
Differences:
- Subject 1: [tex]\(98.7 - 98.2 = 0.5\)[/tex]
- Subject 2: [tex]\(99.7 - 99.1 = 0.6\)[/tex]
- Subject 3: [tex]\(97.5 - 97.3 = 0.2\)[/tex]
- Subject 4: [tex]\(97.3 - 97.6 = -0.3\)[/tex]
- Subject 5: [tex]\(97.7 - 97.4 = 0.3\)[/tex]
2. Calculate the Mean of the Differences ([tex]\(\overline{d}\)[/tex]):
- Add up all the differences and divide by the number of subjects.
[tex]\[
\overline{d} = \frac{0.5 + 0.6 + 0.2 - 0.3 + 0.3}{5} = \frac{1.3}{5} = 0.26
\][/tex]
3. Calculate the Standard Deviation of the Differences ([tex]\(s_d\)[/tex]):
- First, find the squared differences from the mean for each subject.
- Then, find the average of these squared differences.
- Lastly, take the square root of the result to find [tex]\(s_d\)[/tex].
Squared Differences:
- Subject 1: [tex]\((0.5 - 0.26)^2 = 0.0576\)[/tex]
- Subject 2: [tex]\((0.6 - 0.26)^2 = 0.1156\)[/tex]
- Subject 3: [tex]\((0.2 - 0.26)^2 = 0.0036\)[/tex]
- Subject 4: [tex]\((-0.3 - 0.26)^2 = 0.3136\)[/tex]
- Subject 5: [tex]\((0.3 - 0.26)^2 = 0.0016\)[/tex]
Mean of Squared Differences:
[tex]\[
\text{Mean} = \frac{0.0576 + 0.1156 + 0.0036 + 0.3136 + 0.0016}{4} = \frac{0.492}{4} = 0.123
\][/tex]
[tex]\(s_d\)[/tex]:
[tex]\[
s_d = \sqrt{0.123} \approx 0.35
\][/tex]
Thus, the mean of the differences is [tex]\(\overline{d} = 0.26\)[/tex] and the standard deviation is [tex]\(s_d \approx 0.35\)[/tex].
4. Interpretation of [tex]\(\mu_{d}\)[/tex]:
- [tex]\(\mu_{d}\)[/tex] represents the mean population difference between the paired temperatures. It tells us about the average change or difference in temperatures at 8 AM and 12 AM across the entire population from which the samples were drawn.
1. Identify the Paired Differences:
- We have temperatures for five subjects at 8 AM and 12 AM.
- Calculate the difference for each subject by subtracting the 8 AM temperature from the 12 AM temperature.
Differences:
- Subject 1: [tex]\(98.7 - 98.2 = 0.5\)[/tex]
- Subject 2: [tex]\(99.7 - 99.1 = 0.6\)[/tex]
- Subject 3: [tex]\(97.5 - 97.3 = 0.2\)[/tex]
- Subject 4: [tex]\(97.3 - 97.6 = -0.3\)[/tex]
- Subject 5: [tex]\(97.7 - 97.4 = 0.3\)[/tex]
2. Calculate the Mean of the Differences ([tex]\(\overline{d}\)[/tex]):
- Add up all the differences and divide by the number of subjects.
[tex]\[
\overline{d} = \frac{0.5 + 0.6 + 0.2 - 0.3 + 0.3}{5} = \frac{1.3}{5} = 0.26
\][/tex]
3. Calculate the Standard Deviation of the Differences ([tex]\(s_d\)[/tex]):
- First, find the squared differences from the mean for each subject.
- Then, find the average of these squared differences.
- Lastly, take the square root of the result to find [tex]\(s_d\)[/tex].
Squared Differences:
- Subject 1: [tex]\((0.5 - 0.26)^2 = 0.0576\)[/tex]
- Subject 2: [tex]\((0.6 - 0.26)^2 = 0.1156\)[/tex]
- Subject 3: [tex]\((0.2 - 0.26)^2 = 0.0036\)[/tex]
- Subject 4: [tex]\((-0.3 - 0.26)^2 = 0.3136\)[/tex]
- Subject 5: [tex]\((0.3 - 0.26)^2 = 0.0016\)[/tex]
Mean of Squared Differences:
[tex]\[
\text{Mean} = \frac{0.0576 + 0.1156 + 0.0036 + 0.3136 + 0.0016}{4} = \frac{0.492}{4} = 0.123
\][/tex]
[tex]\(s_d\)[/tex]:
[tex]\[
s_d = \sqrt{0.123} \approx 0.35
\][/tex]
Thus, the mean of the differences is [tex]\(\overline{d} = 0.26\)[/tex] and the standard deviation is [tex]\(s_d \approx 0.35\)[/tex].
4. Interpretation of [tex]\(\mu_{d}\)[/tex]:
- [tex]\(\mu_{d}\)[/tex] represents the mean population difference between the paired temperatures. It tells us about the average change or difference in temperatures at 8 AM and 12 AM across the entire population from which the samples were drawn.
Thank you for reading the article Listed below are body temperatures from five different subjects measured at 8 AM and again at 12 AM Find the values of tex overline d. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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