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Answer :
To solve this question, we need to find the average difference in body temperatures at 8 AM and 12 AM, denoted as [tex]$\bar{d}$[/tex], and the standard deviation of these differences, denoted as [tex]$s_d$[/tex]. We'll also explain what [tex]$\mu_d$[/tex] represents.
1. Calculate Differences Between Temperature Measurements:
Subtract each temperature recorded at 8 AM from the corresponding temperature recorded at 12 AM to get the differences.
- Difference for subject 1: 99.1 - 98.3 = 0.8
- Difference for subject 2: 99.1 - 98.6 = 0.5
- Difference for subject 3: 98.0 - 97.8 = 0.2
- Difference for subject 4: 96.8 - 97.3 = -0.5
- Difference for subject 5: 97.9 - 97.6 = 0.3
So, the differences are: 0.8, 0.5, 0.2, -0.5, 0.3.
2. Calculate the Mean of the Differences ([tex]$\bar{d}$[/tex]):
To find the mean difference, add up all the differences and divide by the number of differences.
[tex]\[
\bar{d} = \frac{0.8 + 0.5 + 0.2 + (-0.5) + 0.3}{5} = \frac{1.3}{5} = 0.26
\][/tex]
Therefore, [tex]$\bar{d} = 0.26$[/tex].
3. Calculate the Standard Deviation of Differences ([tex]$s_d$[/tex]):
To find the standard deviation, follow these steps:
a) Find the squared deviations from the mean for each difference:
- For 0.8: [tex]\((0.8 - 0.26)^2\)[/tex]
- For 0.5: [tex]\((0.5 - 0.26)^2\)[/tex]
- For 0.2: [tex]\((0.2 - 0.26)^2\)[/tex]
- For -0.5: [tex]\((-0.5 - 0.26)^2\)[/tex]
- For 0.3: [tex]\((0.3 - 0.26)^2\)[/tex]
b) Calculate the average of these squared deviations (subtract 1 from the count for sample standard deviation, [tex]\( n-1 \)[/tex]):
[tex]\[
s_d = \sqrt{\frac{(0.8 - 0.26)^2 + (0.5 - 0.26)^2 + (0.2 - 0.26)^2 + (-0.5 - 0.26)^2 + (0.3 - 0.26)^2}{5 - 1}}
\][/tex]
After calculation, [tex]\( s_d = 0.4827 \)[/tex].
4. What Does [tex]$\mu_d$[/tex] Represent?
In general, [tex]$\mu_d$[/tex] represents the population mean of the differences in body temperatures at 8 AM and 12 AM. It is the expected value of the differences if we could measure the entire population under the same conditions, rather than just this sample of five subjects.
In conclusion, [tex]$\bar{d}$[/tex] is 0.26, and [tex]$s_d$[/tex] is approximately 0.4827.
1. Calculate Differences Between Temperature Measurements:
Subtract each temperature recorded at 8 AM from the corresponding temperature recorded at 12 AM to get the differences.
- Difference for subject 1: 99.1 - 98.3 = 0.8
- Difference for subject 2: 99.1 - 98.6 = 0.5
- Difference for subject 3: 98.0 - 97.8 = 0.2
- Difference for subject 4: 96.8 - 97.3 = -0.5
- Difference for subject 5: 97.9 - 97.6 = 0.3
So, the differences are: 0.8, 0.5, 0.2, -0.5, 0.3.
2. Calculate the Mean of the Differences ([tex]$\bar{d}$[/tex]):
To find the mean difference, add up all the differences and divide by the number of differences.
[tex]\[
\bar{d} = \frac{0.8 + 0.5 + 0.2 + (-0.5) + 0.3}{5} = \frac{1.3}{5} = 0.26
\][/tex]
Therefore, [tex]$\bar{d} = 0.26$[/tex].
3. Calculate the Standard Deviation of Differences ([tex]$s_d$[/tex]):
To find the standard deviation, follow these steps:
a) Find the squared deviations from the mean for each difference:
- For 0.8: [tex]\((0.8 - 0.26)^2\)[/tex]
- For 0.5: [tex]\((0.5 - 0.26)^2\)[/tex]
- For 0.2: [tex]\((0.2 - 0.26)^2\)[/tex]
- For -0.5: [tex]\((-0.5 - 0.26)^2\)[/tex]
- For 0.3: [tex]\((0.3 - 0.26)^2\)[/tex]
b) Calculate the average of these squared deviations (subtract 1 from the count for sample standard deviation, [tex]\( n-1 \)[/tex]):
[tex]\[
s_d = \sqrt{\frac{(0.8 - 0.26)^2 + (0.5 - 0.26)^2 + (0.2 - 0.26)^2 + (-0.5 - 0.26)^2 + (0.3 - 0.26)^2}{5 - 1}}
\][/tex]
After calculation, [tex]\( s_d = 0.4827 \)[/tex].
4. What Does [tex]$\mu_d$[/tex] Represent?
In general, [tex]$\mu_d$[/tex] represents the population mean of the differences in body temperatures at 8 AM and 12 AM. It is the expected value of the differences if we could measure the entire population under the same conditions, rather than just this sample of five subjects.
In conclusion, [tex]$\bar{d}$[/tex] is 0.26, and [tex]$s_d$[/tex] is approximately 0.4827.
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