20 students from a statistics course were randomly selected to write a test in order to study the influence of video gaming on schoolwork. Their test scores y and the average amount of time they each spent on playing video games per day x were recorded. Fitting a simple linear regression model to the data using command "fit = lm(y x)" in R yields the following output:

> summary(fit)

Call: lm(formula = y ~ x)

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 88.0000 1.6647 52.86 < 2e-16 ***

x -2.9163 0.2268 -12.86 1.64e-10 ***

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Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3.621 on 18 degrees of freedom

Multiple R-squared: 0.9018, Adjusted R-squared: 0.8964

F-statistic: 165.4 on 1 and 18 DF, p-value: 1.643e-10

(1) Give the estimated regression line.

(2) Test for significance of regression at = 0.01 level [must state the null and alternative hypotheses, give the p-value, and state the conclusion; 1 for missing one of these 3 components, 2 for missing two or more of these components; i.e., 0 mark for missing two or more components].

(3) Find the estimate of the error variance ^2 .

(4) find percentage of the total variability in test scores is explained by the linear model

(5) Find the 95% confidence interval for the slope of the model 1. [Some t critical values: t0.025,18 = 2.101, t0.05,18 = 2.552, t0.025,19 = 2.093, t0.05,19 = 2.539, t0.025,20 = 2.086, t0.05,20 = 2.528.]