Answer :
Answer:
ŷ = 7.02895X1 + 12.6959X2 - 170.33728
Since the calculated F=32 falls in the critical region F ≥ 9.55 we reject our null hypothesis and conclude that there is association between at least one of the regressors and the dependent variable.
Step-by-step explanation:
Using stat calculator:
Part A:
x1 x2 y Predicted Y Residual
13 11 62.8739 60.693967 2.179933
15 11 76.1328 74.751867 1.380933
13 13 87.4667 86.085767 1.380933
15 13 102.3236 100.143667 2.179933
14 12 76.1872 80.418817 -4.231617
14 12 77.5287 80.418817 -2.890117
mean
14 12 80.418817 80.418817 0
standard deviation
0.89 0.89 13.2812 12.979739 2.813451
Calculation
Sum of X1 = 84
Sum of X2 = 72
Sum of Y = 482.5129
Mean X1 = 14
Mean X2 = 12
Mean Y = 80.4188
Sum of squares (SSX1) = 4
Sum of squares (SSX2) = 4
Sum of products (SPX1Y) = 28.1158
Sum of products (SPX2Y) = 50.7836
Sum of products (SPX1X2) = 0
Regression Equation = ŷ = b1X1 + b2X2 + a
b1 = ((SPx1y)*(SSx2)-(SPx1x2)*(SPx2y)) / ((SSx1)*(SSx2)-(SPx1x2)*(SPx1x2)) = 112.46/16 = 7.02895
b2 = ((SPx2y)*(SSx1)-(SPx1x2)*(SPx1y)) / ((SSx1)*(SSx2)-(SPx1x2)*(SPx1x2)) = 203.13/16 = 12.6959
a = MY - b1MX1 - b2MX2 = 80.42 - (7.03*14) - (12.7*12) = -170.33728
ŷ = 7.02895X1 + 12.6959X2 - 170.33728
X1-Mx1 X2-Mx2 Y-My (X1-Mx1)² (X2-Mx2)²
-1 -1 -17.545 1 1
1 -1 -4.286 1 1
-1 1 7.048 1 1
1 1 21.905 1 1
0 0 -4.232 0 0
0 0 -2.89 0 0
SSX1: 4 SSX2: 4
SPx1y SPx2y SPx1x2
17.545 17.545 1
-4.286 4.286 -1
-7.048 7.048 - 1
21.905 21.905 1
0 0 0
0 0 0
SPX1Y: SPX2Y: SPX1X2: =0
= 28.116 =50.784
Part B
Coefficient Table
Coefficient SE t- stat
x1 -170.337283 33.519576 -5.081725
x2 7.028950 1.816075 3.870408
b 12.695900 1.816075 6.990847
Part C:
Test for significance of β1 and β2.
State the null and alternate hypotheses as
H0: β1=β2=0
Ha: At least one of the β1 and β2 is non zero.
The significance level is set at ∝= 0.05
The test statistic to use is
F= MSR/ MSE= MS regression/ MS Residual
which if H0 is true has F distribution with υ1= 2 and υ2= n- 3= 6-3= 3 degrees of freedom.
To set up the ANOVA table we find the necessary sum of squares .
Regression SS ( between y^ and y`) = SSR= 842.368060
Residual SS ( between yi and y^) = SSE= 39.577526
Total SS = SSR+ SSE= 842.368060+39.577526= 881.945586
ANOVA table
Source DF Sum of Square Mean Square F Statistic
Regression
(b/w ŷi and yi) 2 842.368060 421.184030 31.926000
Residual
(b/w yi and ŷi) 3 39.577526 13.192509
Total (b/w yi and yi)5 881.945586 176.389117
The critical region is F≥F (0.05) (2,3)=9.55
Conclusion:
Since the calculated F=32 falls in the critical region F ≥ 9.55 we reject our null hypothesis and conclude that there is association between at least one of the regressors and the dependent variable.
Final answer:
The problem requires the application of Multiple Linear Regression to fit a model, estimate standard errors of regression coefficients, and then test the significance of these coefficients using a t-test. However, these calculations ordinarily require the use of a statistical software.
Explanation:
To solve this question, we need to use the method of Multiple Linear Regression. Multiple linear regression aims to model the relationship between two or more independent variables and a dependent variable by fitting a linear equation to observed data.
- First we fit the multiple linear regression model. Typically, this involves using a statistical program to find the line of best fit. This could be the equation y = β0 + β1*x1 + β2*x2.
- Estimating the standard errors usually involves calculating the residuals (the differences between observed and predicted values), squaring them, adding them up, and taking the square root. The formula might look something like this: SE = sqrt( Σ(residuals^2) / n-p ) where n is the number of observations and p is the number of predictors.
- Then we test the significance of the coefficients. This can usually be done using a t-test. The null hypothesis is that the coefficient is equal to zero (meaning it has no effect), and you would reject this hypothesis if the p-value is less than your significance level ( typically 0.05 ).
However, this requires statistical software like SPSS, SAS or R to compute the coefficients values, their standard errors, and the p-values for significance testing.
Learn more about Multiple Linear Regression here:
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