Question

Ten observations on the response variable y associated with two regressor variables x1 and x2 are given in the following table. The model fitted to these observations is yi = β 0 + β 1x1i + β 2x2i + εi , i = 1, …, n , (3) where ε’s are identically and independently distributed as a normal random variable with mean zero and a known variance σ ^2 = 3. Observation # y x1 x2 1 7 9 1 2 8 6 1 3 5 7 1 4 3 8 1 5 2 5 1 6 10 7 -1 7 9 6 -1 8 10 3 -1 9 9 4 -1 10 8 4 -1 a) Test the null hypothesis that there is no difference between the y-intercept for x2 = 1 and the y-intercept for x2 = -1 at a statistical significance level of 0.05. b) Now fit Model (4) to the 10 observations.

Answer 1

a)

The regression equation is defined as,

For X = 1, 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' column then OK. The screenshot is shown below,

The result is obtained. The screenshot is shown below,

The slope coefficient is,

Similarly, the slope coefficient for X = -1 is obtained 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.

Step 3: Select Input Y Range: 'y' column, Input X Range: 'x' column then OK.

The result is obtained. The screenshot is shown below,

The slope coefficient is,

Now, the hypothesis is tested by calcuating t-value and corresponding p-value as shown below,

Null Hypothesis:

Alternate Hypothesis:

The t value is obtained using the formula,

The P-value is obtained using the excel function =T.DIST.2T(x,deg_freedom) where x is the t value degree of freedom = N-1=5-1=4.

The P-value = 0.642007 is greater than 0.05 at 5% significance level. Hence the null hypothesis cannot be rejected. Now we can state that there is a statistically significant evidence that

b)

Now, the multiple 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.

Step 3: Select Input Y Range: 'y' column, Input X Range: 'x1' and 'x2' column then OK.

The result is obtained. The screenshot is shown below,

The regression equation is,