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Confidence interval for beta 1

Suppose you fit the​ first-order multiple regression model y = b0 + b1*x1 + b2*x2 + e
to n = 25 data points and obtain the prediction equation y^ = 28.41 + 2.24x1 + 2*x2.
The estimated standard deviations of the sampling distributions of beta 1
and beta2 are 0.42 and 0.32,  respectively.
a. Test H0: b1 = 0 against Ha: b1 > 0. Use alpha = 0.10

b. Fine a 95% confidence interval for b2. Interpret the interval

The interval above has been constructed for a multiple linear regression model as is always represented by a model with more than one coefficients.  

Y = B0 + B1 + B2 + ... Bz

From this equation, the mode is y = 28.41 + 2.24*x1 + 2*X2

we test the hypothesis that the first coefficient is not significant against the appropriate alternative that the coefficient found is significant. 

To test the slope coefficient, a t statistic for the slope is calculated. 

The formula for the t statistic for slope statistic is; -

t = b1/seb1 , given that the standard deviation for b1 is 0.42, 

t = 2.24/0.42 =  5.333

The value is positive and should be compared with an appropriate critical value for the slope parameter. The degrees of freedom for the slope parameter with 2 predictor variables should be n - 3,  so for this case, the df to be used in checking the critical value should be n-3. 

Use Excel to find the critical value for the slope parameter;-  =T.INV.2T(0.1,23), the critical values are;- -1.714 or 1.714. 

The null hypothesis is rejected in the t statistic for the slope parameter calculated above is not in within -1.714 and 1.714. In this case, reject the null hypothesis, there is enough evidence to conclude that the coefficient parameter is greater than 0. 

Constructing the confidence interval for b2. 

the confidence interval can be calculated using. 

b2 +/- t(alpha/2, df) * Seb2

confidence interval for b2 = 

From Excel;- =T.INV.2T(0.05,23) = 2.069

Thus, 2 +/- 2.069 * 0.32

2 +/- 0.662

The 95% confidence interval for b2 is (1.338, 2.662)


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