Answer :
To solve this problem, we need to find the average difference in body temperatures at 8 AM and 12 AM, as well as the standard deviation of those differences. Additionally, we'll explain what [tex]\(\mu_{d}\)[/tex] represents.
### Step-by-step Solution:
1. Identify the Differences ([tex]\(d\)[/tex]) Between the Two Samples:
For each subject, subtract the temperature at 8 AM from the temperature at 12 AM. These differences ([tex]\(d\)[/tex]) are calculated as follows:
- Subject 1: [tex]\(98.0 - 97.5 = 0.5\)[/tex]
- Subject 2: [tex]\(100.3 - 99.5 = 0.8\)[/tex]
- Subject 3: [tex]\(97.2 - 97.1 = 0.1\)[/tex]
- Subject 4: [tex]\(96.8 - 97.1 = -0.3\)[/tex]
- Subject 5: [tex]\(97.7 - 97.3 = 0.4\)[/tex]
So, the differences [tex]\(d\)[/tex] are: [tex]\(0.5, 0.8, 0.1, -0.3, 0.4\)[/tex].
2. Calculate the Mean of the Differences ([tex]\(\overline{d}\)[/tex]):
The mean of the differences is the sum of all differences divided by the number of subjects.
[tex]\[
\overline{d} = \frac{0.5 + 0.8 + 0.1 - 0.3 + 0.4}{5} = \frac{1.5}{5} = 0.3
\][/tex]
3. Calculate the Standard Deviation of the Differences ([tex]\(s_{d}\)[/tex]):
The standard deviation is a measure of how spread out the differences are. It is calculated using the formula for the sample standard deviation:
[tex]\[
s_{d} = \sqrt{\frac{\sum (d_i - \overline{d})^2}{n-1}}
\][/tex]
Where [tex]\(d_i\)[/tex] is each individual difference, [tex]\(\overline{d}\)[/tex] is the mean of the differences, and [tex]\(n\)[/tex] is the number of differences (in this case, 5).
After carrying out these calculations, the standard deviation [tex]\(s_{d}\)[/tex] is approximately [tex]\(0.4183\)[/tex].
### Conclusion:
- The mean difference [tex]\(\overline{d}\)[/tex] is 0.3.
- The standard deviation [tex]\(s_{d}\)[/tex] is approximately 0.4183.
### Understanding [tex]\(\mu_{d}\)[/tex]:
- [tex]\(\mu_{d}\)[/tex] represents the true mean difference in temperatures between 8 AM and 12 AM for the entire population. While we have calculated [tex]\(\overline{d}\)[/tex] for our sample, [tex]\(\mu_{d}\)[/tex] would be used to denote the average of all possible measurements under similar conditions.
This approach provides you with a thorough understanding and the computed values for [tex]\(\overline{d}\)[/tex] and [tex]\(s_{d}\)[/tex]!
### Step-by-step Solution:
1. Identify the Differences ([tex]\(d\)[/tex]) Between the Two Samples:
For each subject, subtract the temperature at 8 AM from the temperature at 12 AM. These differences ([tex]\(d\)[/tex]) are calculated as follows:
- Subject 1: [tex]\(98.0 - 97.5 = 0.5\)[/tex]
- Subject 2: [tex]\(100.3 - 99.5 = 0.8\)[/tex]
- Subject 3: [tex]\(97.2 - 97.1 = 0.1\)[/tex]
- Subject 4: [tex]\(96.8 - 97.1 = -0.3\)[/tex]
- Subject 5: [tex]\(97.7 - 97.3 = 0.4\)[/tex]
So, the differences [tex]\(d\)[/tex] are: [tex]\(0.5, 0.8, 0.1, -0.3, 0.4\)[/tex].
2. Calculate the Mean of the Differences ([tex]\(\overline{d}\)[/tex]):
The mean of the differences is the sum of all differences divided by the number of subjects.
[tex]\[
\overline{d} = \frac{0.5 + 0.8 + 0.1 - 0.3 + 0.4}{5} = \frac{1.5}{5} = 0.3
\][/tex]
3. Calculate the Standard Deviation of the Differences ([tex]\(s_{d}\)[/tex]):
The standard deviation is a measure of how spread out the differences are. It is calculated using the formula for the sample standard deviation:
[tex]\[
s_{d} = \sqrt{\frac{\sum (d_i - \overline{d})^2}{n-1}}
\][/tex]
Where [tex]\(d_i\)[/tex] is each individual difference, [tex]\(\overline{d}\)[/tex] is the mean of the differences, and [tex]\(n\)[/tex] is the number of differences (in this case, 5).
After carrying out these calculations, the standard deviation [tex]\(s_{d}\)[/tex] is approximately [tex]\(0.4183\)[/tex].
### Conclusion:
- The mean difference [tex]\(\overline{d}\)[/tex] is 0.3.
- The standard deviation [tex]\(s_{d}\)[/tex] is approximately 0.4183.
### Understanding [tex]\(\mu_{d}\)[/tex]:
- [tex]\(\mu_{d}\)[/tex] represents the true mean difference in temperatures between 8 AM and 12 AM for the entire population. While we have calculated [tex]\(\overline{d}\)[/tex] for our sample, [tex]\(\mu_{d}\)[/tex] would be used to denote the average of all possible measurements under similar conditions.
This approach provides you with a thorough understanding and the computed values for [tex]\(\overline{d}\)[/tex] and [tex]\(s_{d}\)[/tex]!