Question

Problem 1: A researcher wishes to see whether a student’s grade-point average and age are related to the student’s score on the state board examination. She selects five students and obtain the following data:

Student	GPA (X1)	Age (X2)	State Board Score (Y)
1	3.2	22	550
2	2.7	27	570
3	2.5	24	525
4	3.4	28	670
5	2.2	23	490

a) Determine the least-squares estimates for the model in which student’s score (Y) is regressed

    on both GPA and Age.

b) Find the estimated model Y on X₁, and X₂

c) Find the student’s predicted score when a student has a GPA of 3.0 and is 25 years old.

d) Test the hypothesis that there is no overall significant regression using all independent

     variables in the model.

e) Report the R²-value and the ANOVA table for the regression Y on X₁, and X₂

Answer 1

a)

the least square estimate of the GPA and Age are,

Explanation:

The regression equation is defined as,

Now, the regression analysis is done in excel by following steps

Step 1: Write the data values in excel. The screenshot is shown below,

Step 2: DATA > Data Analysis > Regression > OK. The screenshot is shown below,

Step 3: Select Input Y Range: 'Y' column, Input X Range: 'X1 and X2' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

From the regression analysis result summary, the least square estimate of the GPA and Age are,

b)

The estimated model Y on X1 and X2 is,

Explanation:

The regression equation is defined as,

From the regression analysis result summary, the least square estimate of the GPA and Age are,

hence the estimated regression model is,

c)

The student’s predicted score = 581.4346

Explanation:

Given: X1 = 3, X2 = 25

From the regression model,

d)

The model is overall statistically significant

Explanation:

The null and alternative hypotheses are defined as,

From the regression model summary,

	Significance F
Regression	0.0213

The significance F value is 0.0213 which is less than 0.05 at a 5% significance level hence the null hypothesis is rejected, which means the model fits the data value at the 5% significance level. Hence we can conclude that the model is overall significant.

e)

R Square

0.9787

Explanation: The R-square value tells, how well the regression model fits the data values. The R-square value of the model is 0.9787 which means, the model explains approximately 97.87% of the variance of the data value. Based on this evidence we can conclude the model is a good fit.