The following table contains the A C T scores and the G P A...

The following table contains the $A C T$ scores and the $G P A$ (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. $$\begin{array}{|ccc|} \hline \text { Student } & G P A & A C T \\ \hline 1 & 2.8 & 21 \\ 2 & 3.4 & 24 \\ 3 & 3.0 & 26 \\ 4 & 3.5 & 27 \\ 5 & 3.6 & 29 \\ 6 & 3.0 & 25 \\ 7 & 2.7 & 25 \\ 8 & 3.7 & 30 \\ \hline \end{array}$$ i. Estimate the relationship between $G P A$ and $A C T$ using $0 \mathrm{LS}$; that is, obtain the intercept and slope estimates in the equation $$\widehat{G P A}=\widehat{\beta}_{0}+\widehat{\beta}_{1} A C T$$ Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the $G P A$ predicted to be if the $A C T$ 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. iii. What is the predicted value of $G P A$ when $A C T=20 ?$ iv. How much of the variation in $G P A$ for these eight students is explained by $A C T$ ? Explain.

The following table contains the $A C T$ scores and the $G P A$ (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.
$$\begin{array}{|ccc|}
\hline \text { Student } & G P A & A C T \\
\hline 1 & 2.8 & 21 \\
2 & 3.4 & 24 \\
3 & 3.0 & 26 \\
4 & 3.5 & 27 \\
5 & 3.6 & 29 \\
6 & 3.0 & 25 \\
7 & 2.7 & 25 \\
8 & 3.7 & 30 \\
\hline
\end{array}$$
i. Estimate the relationship between $G P A$ and $A C T$ using $0 \mathrm{LS}$; that is, obtain the intercept and slope estimates in the equation
$$\widehat{G P A}=\widehat{\beta}_{0}+\widehat{\beta}_{1} A C T$$
Comment on the direction of the relationship. Does the intercept have a useful interpretation here? Explain. How much higher is the $G P A$ predicted to be if the $A C T$ 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.
iii. What is the predicted value of $G P A$ when $A C T=20 ?$
iv. How much of the variation in $G P A$ for these eight students is explained by $A C T$ ? Explain.

Key Concepts

Least Squares Regression

This concept involves finding the line that minimizes the sum of the squared differences between the observed values and the values predicted by the linear model. It’s a fundamental method for estimating the parameters (intercept and slope) in a linear regression model, ensuring the best possible fit to the data in a least-squares sense.

Intercept in Regression

The intercept is the estimated value of the response variable when the predictor variable is zero. It forms a component of the regression line equation, and while it is essential for defining the complete linear model, its interpretation may not always be meaningful if a predictor value of zero is unrealistic or outside the scope of the data.

Slope (Regression Coefficient)

The slope represents the estimated change in the response variable for a one unit change in the predictor variable. It quantifies the strength and direction of the linear relationship between the two variables, indicating whether the relationship is positive or negative.

Fitted Values and Residuals

Fitted values are the predicted responses produced by the regression model for each observation. Residuals are the differences between the actual observed values and the fitted values. Analyzing residuals helps assess the model's adequacy and verify assumptions, such as that residuals sum approximately to zero in linear regression.

Prediction in Regression Analysis

Making predictions involves using the estimated linear model to forecast the response variable for new or specified values of the predictor variable. This concept extends the use of regression models beyond the data used to construct them and is essential for practical applications.

Coefficient of Determination

Also known as R-squared, the coefficient of determination measures the proportion of the variability in the response variable that can be explained by the predictor variable(s) in the model. It provides insight into the model’s explanatory power and goodness-of-fit.

Transcript

00:01 In this part, you are going to show some statistics of the variable net fa.

00:06 This variable indicates the net total financial assets in thousands of dollars.

00:13 This variable has a mean of 1907.

00:20 The standard deviation is 63 .96.

00:29 The minimum level is minus 502 .3 .3 .3 .3.

00:34 And the maximum level is 1 ,536 .8.

00:45 Remember these numbers are in thousand.

00:49 In the second part, you are going to test whether the average net total assets differs by 401k eligibility status, and you will use a two -sided alternative.

01:06 The null hypothesis of this question is the average of net f .a for the that is eligible for 401k.

01:23 Equal net f a of the group that is not eligible for 401k.

01:35 We will do a t test and the t test we will get a value of 13 .1 .1.

01:46 The p value of the test is very small, is way smaller than 0 .001.

01:54 So we are able to reject the non -hypotices that two groups have equal net total financial asset.

02:17 The difference between the two groups in net total financial asset is we will take the value of net total asset, the average, for the group that is eligible for 401k, which is 30 .54.

02:39 These numbers are generated by the statistical software.

02:45 Then we subtract from it the average net fa of the group that is not eligible for 401k, which is 11 .68.

02:59 What we get is roughly 18.

03:05 So the group that is eligible for 401k has a larger average net total financial assets and it is larger by the average of the groups ineligible for 401k by $18 .86 ,000.

03:35 We will estimate a multi -linear regression model for net total financial assets.

03:41 That includes income age eligibility for 401k and we also have age and income included in the regression as quadratic form we will look at the coefficient of the variable e 401k as an indicator of the estimated dollar effect of the eligibility status the coefficient of e401k is 9 .75.

04:27 It means that if the family is eligible for 401k, their net total financial asset will increase by $9 ,705.

04:54 We will add to the model estimated in part 3.

04:59 Interaction terms of eligibility status and h minus 41 and another term is the interaction of eligibility status and h minus 41 square.

05:15 This is the regression result.

05:24 When we look at the interaction terms we see that only this term is significant.

05:33 The term with h minus 41 the term with h -minus 41 square is not significant.

05:53 In future models we can safely drop the last term.

05:57 We will compare the estimated effect of the eligibility status between model in part 3 and model in part 4.

06:07 The left panel show the regression results from the model in part 3 and the right panel show the results from the model in part 4.

06:17 You can tell the difference of the coefficient of e401k between two models.

06:27 In model 3, it is 9 .705.

06:31 In model 4, it is 9 .960.

06:38 You should note that the meaning of the coefficient of e401k differs between two models.

06:48 In model 3, it is the effect for a 4101k.

06:52 All ages in model 4...

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