The following are results from a regression model analysis: ŷ=-101.50+(12.1]34.8 x1+(23.7)5

step 1 First we have to find how many regressors are there in this model. So the given matrix is a symm

The following are results from a regression model analysis:...

The following are results from a regression model analysis: $$ \begin{array}{cc} \hat{y}=-101.50+\underset{(12.1]}{34.8 x_1}+\underset{(23.7)}{56.9 x_2}-\underset{(32.8)}{57.2 x_3} \\ R^2=0.71 & n=65 \end{array} $$ The numbers in parentheses under the coefficients are the estimated coefficient standard errors. a. Compute two-sided $95 \%$ confidence intervals for the three regression slope coefficients. b. For each of the slope coefficients test the hypothesis $$ H_0: \beta_j=0 $$

The following are results from a regression model analysis:
$$
\begin{array}{cc}
\hat{y}=-101.50+\underset{(12.1]}{34.8 x_1}+\underset{(23.7)}{56.9 x_2}-\underset{(32.8)}{57.2 x_3} \\
R^2=0.71 & n=65
\end{array}
$$
The numbers in parentheses under the coefficients are the estimated coefficient standard errors.
a. Compute two-sided $95 \%$ confidence intervals for the three regression slope coefficients.
b. For each of the slope coefficients test the hypothesis
$$
H_0: \beta_j=0
$$

Key Concepts

Confidence Intervals

A confidence interval provides a range of plausible values for a regression parameter based on the data sample. It is calculated using the estimated coefficient along with its standard error and a critical value from the t-distribution corresponding to the desired confidence level. This interval offers an estimated range that, under repeated sampling, would contain the true parameter a specified percentage of the time (e.g., 95%).

Standard Error

The standard error of a regression coefficient quantifies the variability of the estimate due to sample randomness. It is derived from the residual variance and the variability in the predictor. Smaller standard errors suggest more reliable and precise estimates of the regression coefficients.

t-Distribution and Critical Values

The t-distribution is used in the context of small sample sizes or when the population variance is unknown, as is common in regression analysis. The critical values from the t-distribution, at a desired significance level (e.g., 95% confidence), are used to construct confidence intervals and perform hypothesis tests, adjusting for the degrees of freedom available in the analysis.

Hypothesis Testing for Regression Coefficients

Hypothesis testing in regression involves evaluating whether a predictor has a statistically significant relationship with the dependent variable. This is typically done by testing the null hypothesis that the coefficient is equal to zero using a t-test, where the test statistic is calculated as the estimated coefficient divided by its standard error. A statistically significant result indicates that the predictor contributes meaningfully to the model.

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