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Introduction
A student's school performance can be influenced by many factors. A lack of sleep usually causes the student to feel drowsy and be less attentive during classes. I am trying to determine if the number of hours a student get for sleeping per day will significantly determine their test grades. In the Practical Research Methods for Educators by Cipani (link located in green data square below), it shows a raw data table for the test scores of the math midterm and average amount of sleep over a three week period (pg 14). Explanatory variable is the average amount of sleep (hours) over three-week period. Respond variable is the test grade of the students' midterm math exam.
A student's school performance can be influenced by many factors. A lack of sleep usually causes the student to feel drowsy and be less attentive during classes. I am trying to determine if the number of hours a student get for sleeping per day will significantly determine their test grades. In the Practical Research Methods for Educators by Cipani (link located in green data square below), it shows a raw data table for the test scores of the math midterm and average amount of sleep over a three week period (pg 14). Explanatory variable is the average amount of sleep (hours) over three-week period. Respond variable is the test grade of the students' midterm math exam.
Does sleep affect a student's school performance?
Scatterplot: Hours of Sleep vs Test Grades from Math Midterm
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y-axis: Test grade of the students' midterm math exam (%) |
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x-axis: average amount of sleep (hours)
Sample size: 9
Mean x (x̄): 6.6111111111111
Mean y (ȳ): 69.222222222222
Intercept (a): -1.901859504132
Slope (b): 10.75826446281
Regression line equation: y-hat=10.76x-1.90
r=0.94
r^2= 0.89
Sample size: 9
Mean x (x̄): 6.6111111111111
Mean y (ȳ): 69.222222222222
Intercept (a): -1.901859504132
Slope (b): 10.75826446281
Regression line equation: y-hat=10.76x-1.90
r=0.94
r^2= 0.89
Sample: 9 High school students with ADHD
Explanatory Variable: Average amount of sleep (hours) over three-week period
Respond Variable: The test grade of the students' midterm math exam
Slope: 10.76. It shows the rate that the y-values (test grades) are changing for every unit change in the x-value (hours of sleep). It will approximately increase by 10.76% for every hour of sleep a student gets.
y-intercept: -1.9 as the starting point of the least squares regression line. If the number of sleep (hours) is zero, then the regression equation predicts that the test score will be -1.9. However, it is an extrapolation because 0 hours of sleep is not in the range of data so the regression line model will not fit.
2 predicted values using regression line: (7, 73.42) and (5, 51.9). If the regression line is used to determine student's predicted test score with hours of sleep less than 3.75 or greater than 9, there will most likely be an extrapolation because the line is only modeled to fit this set of data.
Overall Pattern
Direction: it appears that the linear regression line has a positive direction, from bottom left to top right. There is a positive association between the hours of sleep and math test scores. (r=0.94)
Form: Along with the positive direction of the data, the scatterplot is in a linear form.
Strength: According to the r value, 0.94, the data has a strong, positive association between hours of sleep and test scores because the correlation coefficient is closer to 1.0.
Outliers: (6, 50) is a possible outlier in this data. It is the most notable point in which it deviates from the linear regression line. If this point was removed, the correlation would be even stronger because the other points would fall closer to the linear pattern. The r and r^2 value would also be increased, but s will decrease since the removal of the outlier gives less variability for the data.
Explanatory Variable: Average amount of sleep (hours) over three-week period
Respond Variable: The test grade of the students' midterm math exam
Slope: 10.76. It shows the rate that the y-values (test grades) are changing for every unit change in the x-value (hours of sleep). It will approximately increase by 10.76% for every hour of sleep a student gets.
y-intercept: -1.9 as the starting point of the least squares regression line. If the number of sleep (hours) is zero, then the regression equation predicts that the test score will be -1.9. However, it is an extrapolation because 0 hours of sleep is not in the range of data so the regression line model will not fit.
2 predicted values using regression line: (7, 73.42) and (5, 51.9). If the regression line is used to determine student's predicted test score with hours of sleep less than 3.75 or greater than 9, there will most likely be an extrapolation because the line is only modeled to fit this set of data.
Overall Pattern
Direction: it appears that the linear regression line has a positive direction, from bottom left to top right. There is a positive association between the hours of sleep and math test scores. (r=0.94)
Form: Along with the positive direction of the data, the scatterplot is in a linear form.
Strength: According to the r value, 0.94, the data has a strong, positive association between hours of sleep and test scores because the correlation coefficient is closer to 1.0.
Outliers: (6, 50) is a possible outlier in this data. It is the most notable point in which it deviates from the linear regression line. If this point was removed, the correlation would be even stronger because the other points would fall closer to the linear pattern. The r and r^2 value would also be increased, but s will decrease since the removal of the outlier gives less variability for the data.
Residual Plot
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y-axis: Residual |
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x-axis: average amount of sleep (hours)
r: 0.94. Since the residual is the difference of the actual y-value and the predicted value that is generated from the linear regression line, the residual shows the strong correlation
r^2: 0.89. The r squared value is 0.89 and shows that the regression line does fit the set of data. The residuals in the residual plot are not too far away from the regression line. The greatest residual point is (6, 50)
s: 5.16, the standard error of the regression and the estimate. The average distance that the observed data points fall from the regression line is about 5.16.
Conclusion
In conclusion, the study has shown that as the hours of sleep increased, the test scores increased as well. The least squares regression line demonstrates that the data has a positive linear correlation between the two quantitative variables (hours of sleep and test scores). There is also a strong relationship between the variables because the correlation coefficient is 0.94, which is very close to 1. There might be lurking variables such as the students' individual IQ score or the time they put into studying that affected the results , so association does not mean causation even if there is a strong positive correlation. Also, the linear regression line is modeled to fit this set of data only, so there would be an extrapolation and error if it is used to determine a student's predicted test score with hours of sleep greater than 9 or less than 3.75. The explanatory variable, average hours of sleep over three week period, did affect the responding variable, the students' test grades. The results generally show that the more hours of sleep a student got, the better he or she did on the math midterm, except for a few exceptions (outlier and lurking variables). Since the experiment only randomly sampled 9 students' scores, it is dangerous to assume that the data applies to all children with ADHD. A limited sample group cannot represent all of the population.
r: 0.94. Since the residual is the difference of the actual y-value and the predicted value that is generated from the linear regression line, the residual shows the strong correlation
r^2: 0.89. The r squared value is 0.89 and shows that the regression line does fit the set of data. The residuals in the residual plot are not too far away from the regression line. The greatest residual point is (6, 50)
s: 5.16, the standard error of the regression and the estimate. The average distance that the observed data points fall from the regression line is about 5.16.
Conclusion
In conclusion, the study has shown that as the hours of sleep increased, the test scores increased as well. The least squares regression line demonstrates that the data has a positive linear correlation between the two quantitative variables (hours of sleep and test scores). There is also a strong relationship between the variables because the correlation coefficient is 0.94, which is very close to 1. There might be lurking variables such as the students' individual IQ score or the time they put into studying that affected the results , so association does not mean causation even if there is a strong positive correlation. Also, the linear regression line is modeled to fit this set of data only, so there would be an extrapolation and error if it is used to determine a student's predicted test score with hours of sleep greater than 9 or less than 3.75. The explanatory variable, average hours of sleep over three week period, did affect the responding variable, the students' test grades. The results generally show that the more hours of sleep a student got, the better he or she did on the math midterm, except for a few exceptions (outlier and lurking variables). Since the experiment only randomly sampled 9 students' scores, it is dangerous to assume that the data applies to all children with ADHD. A limited sample group cannot represent all of the population.
Cipani, Ennio. "Correlation Research Addresses Degree of Association."Practical Research Methods for Educators: Becoming an Evidence-based Practitioner. New York: Springer, 2009. 13-15. Print.
