A researcher usesthe data on class size( ClassSize) and average test scores (Tes

Question

A researcher usesthe data on class size( ClassSize) and average test scores (TestScore) for n = 100 third-grade classes to estimate the model below:

Test Score = 520.4 5.82 Class Size

(20.4) (2.21)

n = 100,R2 = 0.08

(a) Interpret the slope coefficient in this regression. That is, explain what the number 5.82 means.

(b) Construct a 99% confidence interval for the slope coefficient.

(c) A researcher seeks evidence that smaller class sizes provide better learning out- comes for third-graders’. Write down the null and the alternative hypotheses in terms of the slope coefficient for class size (let’s call it 1), and carry out the test at the 5% significance level.

(d) Interpret the R-squared of this regression. That is, explain what R2 = 0.08 means here.

Please solve all parts and show all work

Explanation / Answer

a. The slope of a regression line (b) represents the rate of change in y as x changes. Because y is dependent on x, the slope describes the predicted values of y given x. Here the test score decreases as the class size increases on a scale of 5.82 times.

b. degrees of freedom = 100 =2 = 98.

The critical value is the t score having 98 degrees of freedom and a cumulative probability equal to 0.995. From the t Distribution Calculator, we find that the critical value is 2.627.

ME = critical value * standard error = 2.627 * 2.21 = 5.81.

C.Interval = 5.82 +/- 5.81 = (0.01 ,11.63)

d. R-squared is a statistical measure of how close the data are to the fitted regression line. It is also known as the coefficient of determination, or the coefficient of multiple determination for multiple regression.

R-squared = Explained variation / Total variation

R-squared is always between 0 and 100%:

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