What is the mean difference in the hours of sleep? (Use 4, - 42, where 4, is the amount of sleep South Carolinians had and Hz is the amount of sleep North Carolinians had.) hours Which state, on average slept more? O North Carolina (NC) O South Carolina (Sc) O not enough information O (b) Find the 95% confidence interval for the difference in means of the hours slept.
What is the mean difference in the hours of sleep? (Use 4, - 42, where 4, is the amount of sleep South Carolinians had and Hz is the amount of sleep North Carolinians had.) hours Which state, on average slept more? O North Carolina (NC) O South Carolina (Sc) O not enough information O (b) Find the 95% confidence interval for the difference in means of the hours slept.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Question
![### (a)
**What is the mean difference in the hours of sleep?** (Use μ₁ - μ₂, where μ₁ is the amount of sleep South Carolinians had and μ₂ is the amount of sleep North Carolinians had.)
\[ \_\_\_\_\_\_\_\_\_ \] hours
**Which state, on average slept more?**
- [ ] North Carolina (NC)
- [ ] South Carolina (SC)
- [ ] not enough information
---
### (b)
**Find the 95% confidence interval for the difference in means of the hours slept.**
\[ (\_\_\_\_\_\_\_\_\_, \_\_\_\_\_\_\_\_\_) \]
---
### (c)
**Consider the 90%, 95%, and 99% confidence intervals. Which of the following correctly describes the relationship between the three intervals?**
- [ ] The 90% confidence interval contains both the 95% CI and the 99% CI.
- [ ] The 90% confidence interval is a subset of the 95% CI, which is a subset of the 99% CI.
- [ ] The 95% confidence interval contains both the 90% CI and the 99% CI.
- [ ] The 99% confidence interval is a subset of the 95% CI, which is a subset of the 90% CI.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f11b0a5-f758-48c2-911f-774d5e291a60%2F46f4dba5-8ede-40f1-a1b5-7c7c4c044819%2Fa54wuah_processed.png&w=3840&q=75)
Transcribed Image Text:### (a)
**What is the mean difference in the hours of sleep?** (Use μ₁ - μ₂, where μ₁ is the amount of sleep South Carolinians had and μ₂ is the amount of sleep North Carolinians had.)
\[ \_\_\_\_\_\_\_\_\_ \] hours
**Which state, on average slept more?**
- [ ] North Carolina (NC)
- [ ] South Carolina (SC)
- [ ] not enough information
---
### (b)
**Find the 95% confidence interval for the difference in means of the hours slept.**
\[ (\_\_\_\_\_\_\_\_\_, \_\_\_\_\_\_\_\_\_) \]
---
### (c)
**Consider the 90%, 95%, and 99% confidence intervals. Which of the following correctly describes the relationship between the three intervals?**
- [ ] The 90% confidence interval contains both the 95% CI and the 99% CI.
- [ ] The 90% confidence interval is a subset of the 95% CI, which is a subset of the 99% CI.
- [ ] The 95% confidence interval contains both the 90% CI and the 99% CI.
- [ ] The 99% confidence interval is a subset of the 95% CI, which is a subset of the 90% CI.
![**Oneway Analysis of Hours of Sleep by State**
The image displays a statistical analysis comparing the number of hours of sleep between two states: NC and SC. Various statistical outputs, including box plots and t-tests, are presented.
**Graph: Box Plot of Hours of Sleep by State**
- The box plot illustrates the distribution of sleep hours in NC and SC.
- The y-axis represents the hours of sleep, ranging from 3 to 10 hours.
- Each box shows the interquartile range (IQR), with median values marked.
- Outliers are identified as points outside the whiskers.
**Statistical Summaries and Tests**
1. **Summary of Fit**
- Rsquare: 0.01865 indicates a small proportion of variance explained.
- Root Mean Square Error: 1.183235
- Mean of Response: 7.083333
- Observations: 30
2. **Pooled T Test**
- Difference between SC and NC: 0.3000 hours
- Standard Error of Difference: 0.4357
- 95% Confidence Interval for Difference: [-0.5925, 1.1925]
- Ratio: 0.688535, DF: 28
- Prob > |t|: 0.4988, suggesting no significant difference.
3. **Analysis of Variance (ANOVA)**
- Source: State
- DF (degrees of freedom): 1 for state, 28 for error.
- Sum of Squares: 0.675000 for state, 39.886867 for error.
- Mean Square: 0.675000 for state, 1.42381 for error.
- F Ratio: 0.4741, Prob > F: 0.4968, indicating non-significance.
4. **Means for Oneway ANOVA**
- NC: Mean = 6.9333, Std Error = 0.3089
- SC: Mean = 7.2333, Std Error = 0.3089
- Slight difference in means, but intervals overlap significantly.
5. **T Test Analysis**
- Two side-by-side distributions show the comparison of SC-NC difference, assuming unequal variances.
- 95% Confidence Interval: [-0.4413, 1.0413].
Overall](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f11b0a5-f758-48c2-911f-774d5e291a60%2F46f4dba5-8ede-40f1-a1b5-7c7c4c044819%2Fy0dwlci_processed.png&w=3840&q=75)
Transcribed Image Text:**Oneway Analysis of Hours of Sleep by State**
The image displays a statistical analysis comparing the number of hours of sleep between two states: NC and SC. Various statistical outputs, including box plots and t-tests, are presented.
**Graph: Box Plot of Hours of Sleep by State**
- The box plot illustrates the distribution of sleep hours in NC and SC.
- The y-axis represents the hours of sleep, ranging from 3 to 10 hours.
- Each box shows the interquartile range (IQR), with median values marked.
- Outliers are identified as points outside the whiskers.
**Statistical Summaries and Tests**
1. **Summary of Fit**
- Rsquare: 0.01865 indicates a small proportion of variance explained.
- Root Mean Square Error: 1.183235
- Mean of Response: 7.083333
- Observations: 30
2. **Pooled T Test**
- Difference between SC and NC: 0.3000 hours
- Standard Error of Difference: 0.4357
- 95% Confidence Interval for Difference: [-0.5925, 1.1925]
- Ratio: 0.688535, DF: 28
- Prob > |t|: 0.4988, suggesting no significant difference.
3. **Analysis of Variance (ANOVA)**
- Source: State
- DF (degrees of freedom): 1 for state, 28 for error.
- Sum of Squares: 0.675000 for state, 39.886867 for error.
- Mean Square: 0.675000 for state, 1.42381 for error.
- F Ratio: 0.4741, Prob > F: 0.4968, indicating non-significance.
4. **Means for Oneway ANOVA**
- NC: Mean = 6.9333, Std Error = 0.3089
- SC: Mean = 7.2333, Std Error = 0.3089
- Slight difference in means, but intervals overlap significantly.
5. **T Test Analysis**
- Two side-by-side distributions show the comparison of SC-NC difference, assuming unequal variances.
- 95% Confidence Interval: [-0.4413, 1.0413].
Overall
Expert Solution
Step 1
From the output, the mean difference between the SC(South Carolinians) and NC (North Carolinians) amount of the sleep is 0.30.
Thus, the difference is 0.3 hours
Since, the difference is positive the mean difference of SC is greater than mean difference of NC.
Thus, correct option is
South Carolina (SC)
Step 2
From the data of the confidence level =0.95,
The upper CL diff = 1.1925
The lower CL diff = -0.5925
Thus, 95% confidence interval for the difference in the mean amount of the sleep is (-0.5925 hours, 1.1925 hours)
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