12.4 Suppose you fit a regression model with two independent variables and a sample size of 25. You obtain the least squares equation: Å· = 6.4 + 3.1xâ‚ + 0.92xâ‚‚. The estimated standard deviations of the sampling distributions of β₠and β₂ are 2.3 and 0.27, respectively. A.) Test Hâ‚€: β₠= 0 against the alternative Hâ‚: β₠> 0. Use α = 0.05. B.) Test Hâ‚€: β₂ = 0 against the alternative Hâ‚: β₂ ≠0. Use α = 0.05. C.) Find a 90% confidence interval for β₠and interpret. D.) Find a 99% confidence interval for β₂ and interpret this interval.
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) To test \( H_0: \beta_1 = 0 \) against \( H_a: \beta_1 > 0 \) with \( \alpha = 0.05 \), we first calculate the t-statistic for \( \beta_1 \) using the formula: \[ t = \frac{\hat{\beta}_1 - \beta_{1,0}}{SE(\hat{\beta}_1)} \] Given \( \hat{\beta}_1 = 3.1 \) and Show more…
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MINITAB was used to fit the model $y=\beta_{0}+\beta_{1} x_{1}+$ $\beta_{2} x_{2}+\varepsilon$ to $n=20$ data points, and the printout (top of page 628 ) was obtained. a. What are the sample estimates of $\beta_{0}, \beta_{1},$ and $\beta_{2}$ ? b. What is the least squares prediction equation? c. Find SSE, MSE, and $s$. Interpret the standard deviation in the context of the problem. d. Test $H_{0}: \beta_{1}=0$ against $H_{a}: \beta_{1} \neq 0 .$ Use $\alpha=.05$. e. Use a $95 \%$ confidence interval to estimate $\beta_{2}$. f. Find $R^{2}$ and $R_{t}^{2}$ and interpret these values. g. Use the two formulas given in this section to calculate the test statistic for the null hypothesis $H_{0}-\beta_{1}=\beta_{2}=0$.
Use the properties of the least-squares estimators given in Section 11.4 to complete the following. a. Show that under the null hypothesis $H_{0}: \beta_{i}=\beta_{i 0}$ $$T=\frac{\widehat{\beta}_{i}-\beta_{i 0}}{S \sqrt{c_{i i}}}$$ possesses a $t$ distribution with $n-2$ df, where $i=1,2$. b. Derive the confidence intervals for $\beta_{i}$ given in this section.
Linear Models and Estimation by Least Squares
Inferences Concerning the Parameters $\beta_{i}$
You estimate a simple linear regression model using a sample of 25 observations and obtain the following results (estimated standard errors in parentheses below coefficient estimates): y = 97.25 + 19.74 * x (3.86) (3.42) You want to test the following hypothesis: H0: β2 = 1, H1: β2 > 1. If you choose to reject the null hypothesis based on these results, what is the probability you have committed a Type I error? a.) between .01 and .02 b.) between .02 and .05 c.) less than .005 d.) It is impossible to determine without knowing the true value of β2
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