Question

Exercise 11.2 A psychologist speculates that spending a lot of time on the internet has a negative effect on children's sleep. Consider the following data on hours of deep sleep (Y) and hours spent on the internet (X) where $x_i$ and $y_i$ are the observations on internet time and deep sleep time of the $i^{th}$ ($i = 1, 2, ..., 9$) child respectively: egin{tabular}{|c|c|c|c|c|c|c|c|c|c|} hline Child $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \ hline Internet time $x_i$ (in h) & 0.3 & 2.2 & 0.5 & 0.7 & 1.0 & 1.8 & 3.0 & 0.2 & 2.3 \ Sleep time $y_i$ (in h) & 5.8 & 4.4 & 6.5 & 5.8 & 5.6 & 5.0 & 4.8 & 6.0 & 6.1 \ hline end{tabular} (a) Estimate the linear regression model for the given data and interpret the coeffi- cients. (b) Calculate $R^2$ to judge the goodness of fit of the model. (c) Reproduce the results of a) and b) in R and plot the regression line. (d) Now assume that we only distinguish between spending more than 1 hour on the internet ($X = 1$) and spending less than (or equal to) one hour on the internet ($X = 0$). Estimate the linear model again and compare the results. How can $hat{eta}$ now be interpreted? Describe how $hat{eta}$ changes if those who spend more than one hour on the internet are coded as 0 and the others as 1.

          Exercise 11.2 A psychologist speculates that spending a lot of time on the internet
has a negative effect on children's sleep. Consider the following data on hours of deep
sleep (Y) and hours spent on the internet (X) where $x_i$ and $y_i$ are the observations
on internet time and deep sleep time of the $i^{th}$ ($i = 1, 2, ..., 9$) child respectively:

egin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
hline
Child $i$ & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \
hline
Internet time $x_i$ (in h) & 0.3 & 2.2 & 0.5 & 0.7 & 1.0 & 1.8 & 3.0 & 0.2 & 2.3 \
Sleep time $y_i$ (in h) & 5.8 & 4.4 & 6.5 & 5.8 & 5.6 & 5.0 & 4.8 & 6.0 & 6.1 \
hline
end{tabular}

(a) Estimate the linear regression model for the given data and interpret the coeffi-
cients.
(b) Calculate $R^2$ to judge the goodness of fit of the model.
(c) Reproduce the results of a) and b) in R and plot the regression line.
(d) Now assume that we only distinguish between spending more than 1 hour on the
internet ($X = 1$) and spending less than (or equal to) one hour on the internet
($X = 0$). Estimate the linear model again and compare the results. How can $hat{eta}$ 
now be interpreted? Describe how $hat{eta}$ changes if those who spend more than one
hour on the internet are coded as 0 and the others as 1.

Exercise 11.2 A psychologist speculates that spending a lot of time on the internet
has a negative effect on children's sleep. Consider the following data on hours of deep
sleep (Y) and hours spent on the internet (X) where xi and yi are the observations
on internet time and deep sleep time of the i^th (i = 1, 2, ..., 9) child respectively:

egintabular|c|c|c|c|c|c|c|c|c|c|
hline
Child i 1 2 3 4 5 6 7 8 9 hline
Internet time xi (in h) 0.3 2.2 0.5 0.7 1.0 1.8 3.0 0.2 2.3 Sleep time yi (in h) 5.8 4.4 6.5 5.8 5.6 5.0 4.8 6.0 6.1 hline
endtabular

(a) Estimate the linear regression model for the given data and interpret the coeffi-
cients.
(b) Calculate R^2 to judge the goodness of fit of the model.
(c) Reproduce the results of a) and b) in R and plot the regression line.
(d) Now assume that we only distinguish between spending more than 1 hour on the
internet (X = 1) and spending less than (or equal to) one hour on the internet
(X = 0). Estimate the linear model again and compare the results. How can hateta
now be interpreted? Describe how hateta changes if those who spend more than one
hour on the internet are coded as 0 and the others as 1.

Added by Mckenzie H.

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