3 The following table contains the ACT scores and the GPA...

3 The following table contains the ACT scores and the GPA (grade point average) for eight college students. Grade point average is based on a four-point scale and has been rounded to one digit after the decimal. Student GPA ACT 1 2.8 21 2 3.4 24 3 3.0 26 4 3.5 27 5 3.6 29 6 3.0 25 7 2.7 25 8 3.7 30 (i) Estimate the relationship between GPA and ACT using OLS; that is, obtain the intercept and slope estimates in the equation $widehat{GPA} = eta_0 + eta_1 ACT$. Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the GPA predicted to be if the ACT score is increased by five points? (ii) Compute the fitted values and residuals for each observation, and verify that the residuals (approximately) sum to zero.

Transcript

00:01 We're looking at the association between gpa and act scores.

00:04 And act scores are going to be our independent variable and gpa will be the dependent variable.

00:11 And so we're going to make a least scores regression line where we have the gpa.

00:18 This will be like the y, this is the gpa.

00:24 It's going to be y hats or predictor, it's a predicted value, is equal to beta not plus beta sub 1 times the act score, the x, which is the act score.

00:37 All right, so we want to figure out these values.

00:40 So beta 1, the slope coefficient, is calculated as r times the standard deviation of y, divided by the standard deviation of x.

00:50 Then beta not, the intercept is equal to y bar, the mean of the y values, minus b, the beta 1 times x bar, the mean of the x's.

01:05 So let's go and do this.

01:08 So i use these spreadsheet functions.

01:10 I'll show you what i got here.

01:14 I'll show you the whole things.

01:16 These are what they are.

01:16 I didn't round.

01:17 I let the spreadsheet do all the work for us.

01:20 So to calculate the means, the averages, the these values, i use the average function, average, putting the data, and out pops the mean.

01:31 And then for the sample standard deviation, make sure you do the sample standard deviation.

01:35 It's st -d -e -v -d -s.

01:38 And again, putting your data.

01:39 Just select this whole cell reference and then you get these values here for the y and x variables for the gpa and the act scores respectively.

01:48 And then to get the correlation coefficient r, i use this function called corel.

01:54 And for that what you do is you put in the y values and then the x values, like in terms of the cell references.

02:02 You select y's comma, x is common, and then you get that.

02:08 And then we're asked to do it later on, but we'll do our right now is to get the r squared value because we're asked to state how much variation in the y variable in the gpa that can be explained by the x variable and it literally is the r value squared.

02:30 So it's 57 .7 % of the variation we see in gpa is explained by the variation we see in the act scores.

02:38 All right so let's keep moving along here.

02:43 Let's interpret the coefficients here.

02:49 Let's see.

02:49 We calculated them here.

02:50 Here's the beta one and beta not values.

02:53 So let's put our whole function together.

02:55 Y hat gpa equals the beta not 0 .5, 6...

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