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8.5 Confidence intervals for the regression coefficients

We can also use the sampling distributions of $\hat{\beta}_{j}$ and $\hat{\sigma}^{2}$ to create a $100(1-\alpha)\%$ confidence interval for $\beta_{j}$ , {mdframed}

\displaystyle\hat{\beta}_{j}\pm t_{n-p}(1-\alpha/2)\times\sqrt{\hat{\sigma}^{2% }(X^{\prime}X)^{-1}_{j,j}}

As discussed in Part 1, the confidence interval can be used to test $H_{0}:\beta_{j}=b$ against $H_{1}:\beta_{j}\neq b$ . The null hypothesis is rejected at the $\alpha\%$ significance level if $b$ does not lie in the $100(1-\alpha)\%$ confidence interval.

To test against the one-tailed alternatives,

1

$H_{1}:\beta_{j}>b$ . Calculate the $100(1-2\alpha)\%$ confidence interval and reject $H_{0}$ at the $\alpha\%$ level if $b$ lies below the lower bound of the confidence interval;
2

$H_{1}:\beta_{j}<b$ . Calculate the $100(1-2\alpha)\%$ confidence interval and reject $H_{0}$ at the $\alpha\%$ level if $b$ lies above the upper bound of the confidence interval.

TheoremExample 8.5.1 Birth weights: confidence interval and two tailed test

Derive a 95% confidence interval for the regression coefficient representing this relationship between weight and gestational age at birth.

We have all the information to do this from the previous example,

1

$\hat{\beta}_{2}=116$ , $\operatorname{se}(\hat{\beta}_{2})=\sqrt{\hat{\sigma}^{2}(X^{\prime}X)^{-1}_{2% ,2}}=22.2$ and $t_{22}(0.975)=2.074$ .
2

Then the 95% confidence interval for $\beta_{2}$ is

$\displaystyle\hat{\beta}_{2}\pm t_{22}(0.975)\times\sqrt{\hat{\sigma}^{2}(X^{% \prime}X)^{-1}_{2,2}}=116\pm 2.074\times 22.2=(70.0,162.0).$

Since zero lies outside this interval, there is evidence at the 5% level to reject $H_{0}$ , i.e. there is evidence of a relationship between gestational age and weight at birth.

TheoremExample 8.5.2 Birth weights: confidence interval and one-tailed test

Since zero lies below the confidence interval, we might want to test

\displaystyle H_{0}:\beta_{2}=0

vs.

\displaystyle H_{1}:\beta_{2}>0.

To test at the 5% level, use $t_{22}(0.95)=1.717$ to calculate a 90% confidence interval and see if zero lies to the left of this confidence interval.

As above,

\displaystyle\hat{\beta}_{2}\pm t_{22}(0.95)\times\sqrt{\hat{\sigma}^{2}(X^{% \prime}X)^{-1}_{2,2}}=116\pm 1.717\times 22.2=(77.9,154.1)

Since $0<77.9$ , we conclude that there is evidence for a positive relationship between gestational age and weight at birth.

8.6 Summary