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 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].
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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