Answer :
To solve this problem, we'll calculate the average difference, [tex]\(\overline{d}\)[/tex], and the standard deviation of the differences, [tex]\(s_d\)[/tex], between body temperatures measured at two different times (8 AM and 12 AM) for five subjects.
Here are the steps:
1. List the Temperatures:
- At 8 AM: 98.3, 99.5, 97.7, 97.2, 97.4
- At 12 AM: 98.8, 99.8, 98.0, 97.0, 97.5
2. Calculate the Differences:
Find the difference for each subject by subtracting the 8 AM measurement from the 12 AM measurement:
- [tex]\(98.8 - 98.3 = 0.5\)[/tex]
- [tex]\(99.8 - 99.5 = 0.3\)[/tex]
- [tex]\(98.0 - 97.7 = 0.3\)[/tex]
- [tex]\(97.0 - 97.2 = -0.2\)[/tex]
- [tex]\(97.5 - 97.4 = 0.1\)[/tex]
So, the differences are: 0.5, 0.3, 0.3, -0.2, 0.1.
3. Calculate the Mean Difference ([tex]\(\overline{d}\)[/tex]):
To find [tex]\(\overline{d}\)[/tex], compute the average of the differences:
[tex]\[
\overline{d} = \frac{0.5 + 0.3 + 0.3 - 0.2 + 0.1}{5} = \frac{1.0}{5} = 0.2
\][/tex]
4. Calculate the Standard Deviation of the Differences ([tex]\(s_d\)[/tex]):
To find [tex]\(s_d\)[/tex], first find the variance by calculating each squared difference from the mean difference, then average these squared differences, and finally take the square root.
1. Differences from the mean:
- [tex]\(0.5 - 0.2 = 0.3\)[/tex]
- [tex]\(0.3 - 0.2 = 0.1\)[/tex]
- [tex]\(0.3 - 0.2 = 0.1\)[/tex]
- [tex]\(-0.2 - 0.2 = -0.4\)[/tex]
- [tex]\(0.1 - 0.2 = -0.1\)[/tex]
2. Squared differences:
- [tex]\(0.3^2 = 0.09\)[/tex]
- [tex]\(0.1^2 = 0.01\)[/tex]
- [tex]\(0.1^2 = 0.01\)[/tex]
- [tex]\((-0.4)^2 = 0.16\)[/tex]
- [tex]\((-0.1)^2 = 0.01\)[/tex]
3. Average squared difference (variance) using sample formula (divide by 4 since it's a sample for standard deviation):
[tex]\[
\text{variance} = \frac{0.09 + 0.01 + 0.01 + 0.16 + 0.01}{4} = \frac{0.28}{4} = 0.07
\][/tex]
4. Standard deviation:
[tex]\[
s_d = \sqrt{0.07} \approx 0.2646
\][/tex]
5. Interpret [tex]\(\mu_d\)[/tex]:
The symbol [tex]\(\mu_d\)[/tex] represents the population mean of the differences. It indicates the average difference would expect if you measured the entire population under similar conditions instead of a sample.
Therefore, the values are [tex]\(\overline{d} = 0.2\)[/tex] and [tex]\(s_d \approx 0.2646\)[/tex].
Here are the steps:
1. List the Temperatures:
- At 8 AM: 98.3, 99.5, 97.7, 97.2, 97.4
- At 12 AM: 98.8, 99.8, 98.0, 97.0, 97.5
2. Calculate the Differences:
Find the difference for each subject by subtracting the 8 AM measurement from the 12 AM measurement:
- [tex]\(98.8 - 98.3 = 0.5\)[/tex]
- [tex]\(99.8 - 99.5 = 0.3\)[/tex]
- [tex]\(98.0 - 97.7 = 0.3\)[/tex]
- [tex]\(97.0 - 97.2 = -0.2\)[/tex]
- [tex]\(97.5 - 97.4 = 0.1\)[/tex]
So, the differences are: 0.5, 0.3, 0.3, -0.2, 0.1.
3. Calculate the Mean Difference ([tex]\(\overline{d}\)[/tex]):
To find [tex]\(\overline{d}\)[/tex], compute the average of the differences:
[tex]\[
\overline{d} = \frac{0.5 + 0.3 + 0.3 - 0.2 + 0.1}{5} = \frac{1.0}{5} = 0.2
\][/tex]
4. Calculate the Standard Deviation of the Differences ([tex]\(s_d\)[/tex]):
To find [tex]\(s_d\)[/tex], first find the variance by calculating each squared difference from the mean difference, then average these squared differences, and finally take the square root.
1. Differences from the mean:
- [tex]\(0.5 - 0.2 = 0.3\)[/tex]
- [tex]\(0.3 - 0.2 = 0.1\)[/tex]
- [tex]\(0.3 - 0.2 = 0.1\)[/tex]
- [tex]\(-0.2 - 0.2 = -0.4\)[/tex]
- [tex]\(0.1 - 0.2 = -0.1\)[/tex]
2. Squared differences:
- [tex]\(0.3^2 = 0.09\)[/tex]
- [tex]\(0.1^2 = 0.01\)[/tex]
- [tex]\(0.1^2 = 0.01\)[/tex]
- [tex]\((-0.4)^2 = 0.16\)[/tex]
- [tex]\((-0.1)^2 = 0.01\)[/tex]
3. Average squared difference (variance) using sample formula (divide by 4 since it's a sample for standard deviation):
[tex]\[
\text{variance} = \frac{0.09 + 0.01 + 0.01 + 0.16 + 0.01}{4} = \frac{0.28}{4} = 0.07
\][/tex]
4. Standard deviation:
[tex]\[
s_d = \sqrt{0.07} \approx 0.2646
\][/tex]
5. Interpret [tex]\(\mu_d\)[/tex]:
The symbol [tex]\(\mu_d\)[/tex] represents the population mean of the differences. It indicates the average difference would expect if you measured the entire population under similar conditions instead of a sample.
Therefore, the values are [tex]\(\overline{d} = 0.2\)[/tex] and [tex]\(s_d \approx 0.2646\)[/tex].