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Write the formula for the estimated regression line and interpret the slope of the estimated regression line, the intercept of the estimated regression line- Is it meaningful?, and the estimated R2. Based on the fitted regression model, what is the predicted ATST for a child who is 7 years old? What is the correlation between AGE and ATST?

Write the formula for the estimated regression line and interpret the slope of the estimated regression line, the intercept of the estimated regression line- Is it meaningful?, and the estimated R2. Based on the fitted regression model, what is the predicted ATST for a child who is 7 years old? What is the correlation between AGE and ATST?

MATLAB: An Introduction with Applications
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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  • Write the formula for the estimated regression line and interpret the slope of the estimated regression line, the intercept of the estimated regression line- Is it meaningful?, and the estimated R2.
  • Based on the fitted regression model, what is the predicted ATST for a child who is 7 years old?
  • What is the correlation between AGE and ATST?
  • Does the residual plot suggest that the fitted regression line is inappropriate for these data? Explain why or why not.

 

  • Suppose that a new subject is added to the study data and that subject is 12.5 years old with an ATST of 580 minutes. If the regression model were to be refit with this additional data point, would the new slope be greater than or less than -14.041?  Justify your response.
**Study on Age and Average Total Sleep Time (ATST)** **Overview:** A study involving 13 healthy children and adolescents aimed to examine the relationship between age (measured in years) and average total sleep time (ATST, measured in minutes). For accuracy, ATST was determined by recording sleep time over three consecutive nights and then averaging the results. **Data Table:** | Subject | Age (years) | ATST (minutes) | |---------|-------------|----------------| | 1 | 4.40 | 586.00 | | 2 | 14.00 | 461.75 | | 3 | 10.10 | 491.10 | | 4 | 6.70 | 565.00 | | 5 | 11.50 | 462.00 | | 6 | 9.60 | 532.10 | | 7 | 12.40 | 477.60 | | 8 | 8.90 | 515.20 | | 9 | 11.10 | 493.00 | | 10 | 7.75 | 528.30 | | 11 | 5.50 | 575.90 | | 12 | 8.60 | 532.50 | | 13 | 7.20 | 530.50 | **Scatter Plot Explanation:** - **Axes**: - The x-axis represents age in years. - The y-axis represents average total sleep time in minutes. - **Data Points**: - Each point on the scatter plot corresponds to one subject's age and their averaged total sleep time. - **Trend**: - The scatter plot shows a clear downward trend, suggesting that as age increases, the average total sleep time decreases. This information could be useful for understanding patterns in sleep needs relative to age in children and adolescents.
Transcribed Image Text:**Study on Age and Average Total Sleep Time (ATST)** **Overview:** A study involving 13 healthy children and adolescents aimed to examine the relationship between age (measured in years) and average total sleep time (ATST, measured in minutes). For accuracy, ATST was determined by recording sleep time over three consecutive nights and then averaging the results. **Data Table:** | Subject | Age (years) | ATST (minutes) | |---------|-------------|----------------| | 1 | 4.40 | 586.00 | | 2 | 14.00 | 461.75 | | 3 | 10.10 | 491.10 | | 4 | 6.70 | 565.00 | | 5 | 11.50 | 462.00 | | 6 | 9.60 | 532.10 | | 7 | 12.40 | 477.60 | | 8 | 8.90 | 515.20 | | 9 | 11.10 | 493.00 | | 10 | 7.75 | 528.30 | | 11 | 5.50 | 575.90 | | 12 | 8.60 | 532.50 | | 13 | 7.20 | 530.50 | **Scatter Plot Explanation:** - **Axes**: - The x-axis represents age in years. - The y-axis represents average total sleep time in minutes. - **Data Points**: - Each point on the scatter plot corresponds to one subject's age and their averaged total sleep time. - **Trend**: - The scatter plot shows a clear downward trend, suggesting that as age increases, the average total sleep time decreases. This information could be useful for understanding patterns in sleep needs relative to age in children and adolescents.
### Simple Linear Regression Analysis A simple linear regression model was fit to the data in R. The results are given below: #### Model Call: ```r lm(formula = ATST ~ AGE) ``` #### Residuals: - **Min:** -23.011 - **1Q:** -9.365 - **Median:** 2.372 - **3Q:** 6.770 - **Max:** 20.411 #### Coefficients: | | Estimate | Std. Error | t value | Pr(>|t|) | |--------------|----------|------------|--------|-------------| | (Intercept) | 646.483 | 12.918 | 50.05 | 2.49e-14 *** | | AGE | -14.041 | 1.368 | -10.26 | 5.70e-07 *** | **Significance codes:** - 0 ‘***’ - 0.001 ‘**’ - 0.01 ‘*’ - 0.05 ‘.’ - 0.1 ‘ ’ - 1 #### Additional Statistics: - **Residual standard error:** 13.15 on 11 degrees of freedom - **Multiple R-squared:** 0.9054 - **Adjusted R-squared:** 0.8968 - **F-statistic:** 105.3 on 1 and 11 DF - **p-value:** 5.7e-07 --- ### Graph Explanation Below is a plot of the model residuals versus age. The x-axis represents age in years, and the y-axis represents the residuals. - The chart is a scatter plot with residuals ranging from approximately -20 to 20. - The points do not exhibit a clear pattern, which is generally desirable, indicating that the model's assumptions about the data are reasonable.
Transcribed Image Text:### Simple Linear Regression Analysis A simple linear regression model was fit to the data in R. The results are given below: #### Model Call: ```r lm(formula = ATST ~ AGE) ``` #### Residuals: - **Min:** -23.011 - **1Q:** -9.365 - **Median:** 2.372 - **3Q:** 6.770 - **Max:** 20.411 #### Coefficients: | | Estimate | Std. Error | t value | Pr(>|t|) | |--------------|----------|------------|--------|-------------| | (Intercept) | 646.483 | 12.918 | 50.05 | 2.49e-14 *** | | AGE | -14.041 | 1.368 | -10.26 | 5.70e-07 *** | **Significance codes:** - 0 ‘***’ - 0.001 ‘**’ - 0.01 ‘*’ - 0.05 ‘.’ - 0.1 ‘ ’ - 1 #### Additional Statistics: - **Residual standard error:** 13.15 on 11 degrees of freedom - **Multiple R-squared:** 0.9054 - **Adjusted R-squared:** 0.8968 - **F-statistic:** 105.3 on 1 and 11 DF - **p-value:** 5.7e-07 --- ### Graph Explanation Below is a plot of the model residuals versus age. The x-axis represents age in years, and the y-axis represents the residuals. - The chart is a scatter plot with residuals ranging from approximately -20 to 20. - The points do not exhibit a clear pattern, which is generally desirable, indicating that the model's assumptions about the data are reasonable.
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