GPA vs. Hours of Sleep: A Simple Linear Regression Jacob Ushkurnis 12/16/2016

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1 GPA vs. Hours of Sleep: A Simple Linear Regression Jacob Ushkurnis 12/16/2016 Introduction As a college student, life can sometimes get extremely busy and stressful when there is a lot of work to do. More often than not, homework assignments can cause students to stay up very late at night, and therefore impact how much sleep they are able to get. An important question to then ask and investigate is, Does the average amount of sleep college students get impact their GPA at all? It may be assumed that students who get more sleep are more likely to have higher GPA s because they are more well-rested, but this also may not be the case if students who get more sleep are simply getting more sleep because they are not doing their work. Methods This study was conducted using a survey that was handed out to 50 random people. The 50 random subjects were given the option to take the survey or not before a couple of my classes started. Random individuals were chosen to participate in this study rather than just 50 of my friends because it was desired to have the most random data possible, and to decrease the chances of selection bias. The subjects filled out the printed surveys and then I collected them at the end of class. Then, the data was input into Microsoft Excel. The topic that is being analyzed is whether or not there is an association between hours of sleep and GPA, based off of the data from 50 randomly selected college students from the University of Massachusetts, Amherst. The inference procedure being used is a simple linear regression. The independent variable (x) in this case is hours of sleep, and the dependent variable (y) is GPA. The alpha value that will be used in order to either accept/reject the coefficients found by the model is 0.05 (95% Confidence). Judging by the adjusted R-squared value, p-values, and correlation coefficients, tentative conclusions will be made regarding the association (or lack thereof) relating hours of sleep to GPA. Results Preliminary Analysis Numerical Descriptions of Data library(ggplot2) setwd("/users/jacob_ushkurnis/downloads") mydata <- read.csv("sleep_vs_gpa_data.csv") attach(mydata) summary(mydata) sleep gpa Min. : 4 Min. : st Qu.: 6 1st Qu.:3.455 Median : 7 Median :3.600 Mean : 7 Mean :

2 3rd Qu.: 8 3rd Qu.:3.715 Max. :10 Max. :4.000 sd(sleep) [1] sd(gpa) [1] Graphical Descriptions of Data qplot(mydata$sleep, geom="histogram", main="histogram for Average Hours of Sleep", xlab="average Hours of Sleep", ylab="count") 10.0 Histogram for Average Hours of Sleep 7.5 Count Average Hours of Sleep 2

3 qplot(mydata$gpa, geom="histogram", main="histogram for GPA", xlab="gpa", ylab="count") 4 Histogram for GPA 3 Count GPA 3

4 boxplot(gpa~sleep,data=mydata, main="boxplots for GPA vs. Average Hours of Sleep", xlab="average Hours o Boxplots for GPA vs. Average Hours of Sleep GPA Average Hours of Sleep 4

5 Formal Analysis Plot of GPA vs. Average Hours of Sleep, plus Regression Line ggplot(mydata, aes(x=sleep, y=gpa)) + geom_point(shape=1) + geom_smooth(method=lm)+ ggtitle("gpa vs. Average Hours of Sleep")+ labs(x="average Hours of Sleep", y="gpa") 4.00 GPA vs. Average Hours of Sleep 3.75 GPA Average Hours of Sleep 5

6 Regression Summary lm1 <- lm(gpa~sleep) summary(lm1) Call: lm(formula = gpa ~ sleep) Residuals: Min 1Q Median 3Q Max Coefficients: Estimate Std. Error t value Pr(> t ) (Intercept) < 2e-16 *** sleep e-05 *** --- Signif. codes: 0 '***' '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: on 48 degrees of freedom Multiple R-squared: , Adjusted R-squared: F-statistic: on 1 and 48 DF, p-value: 2.752e-05 Confidence Interval for Sleep confint(lm1, 'sleep', level=0.95) 2.5 % 97.5 % sleep Conclusions With an adjusted R-squared value of , it can be concluded that about 29.49% of the variation in GPA can be explained by this model. Additionally, it can be seen that there is a relatively weak, positive correlation between average hours of sleep and GPA. The p-value of both the intercept and of the sleep variable are both far below 0.05, therefore they are statistically significant. The interpetation of the slope can be described as: If the average hours of sleep a student gets increases by 1 hour, then the we can expect (on average) a point increase in GPA. The interpetation of the intercept can be described as: If the average hours of sleep someone gets is zero (they do not sleep), then we can expect their GPA to be approximately With these data, the intercept does not quite make sense. It is not plausible for someone to get zero hours of sleep (no sleep at all). Additionally, it is important to note that the linear model will not make logical sense in terms of GPA (out of a 4.0 scale) for x-values greater than In other words, if a student gets hours of sleep per night on average or more, their predicted GPA from this model will be greater than 4.0, which is not possible. 6