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UNSW MATH2089 Sample 1 Q3.tex
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\documentclass[10pt]{article} \usepackage{amssymb,amsmath} \usepackage[hmargin=1cm,vmargin=1cm]{geometry} \usepackage{cancel} \begin{document} {\large UNSW MATH2089 Sample 1 Q3} \begin{align*} \text{Q3 }&\text{Concrete is vulnerable to shock vibrations, which may cause hidden damage}\\ &\text{to the material. In a study of vibration phenomena, an experiment is carried}\\ &\text{out and data is reported: including the variables {\bf ppv} -- peak particle velocity}\\ &\text{(mm/sec), and {\bf Ratio} -- ratio of ultrasonic pulse velocity after impact to that}\\ &\text{before impact in concrete prisms. Investigators fit the simple linear linear}\\ &\text{regression model:}\\ \\ &\bf Ratio=\beta_1+\beta_2\times ppv+\epsilon\qquad(*)\\ &\text{Note: The question uses $\beta_1$ and $\beta_2$, instead of $\beta_0$ and $\beta_1$ as in the lecture.}\\ \\ &\text{Using the regression analysis output (which includes residual plots and a fitted}\\ &\text{line plot) at the end of the question, answer the following questions.}\\ \\ \text{a) i) }&\text{Write the null and alternative hypotheses to test whether the variable}\\ &\text{{\bf ppv} is significant in predicting the variable {\bf Ratio}.}\\ \\ &\text{The null hypotheses }H_0:\beta_2=0\qquad\text{against}\\ &\text{The alternative hypothese }H_a:\beta_2\le 0.\\ \\ \text{ii) }&\text{Carry out the test at the 1\% significance level.}\\ \\ &\alpha=0.01,\quad 1-\frac{\alpha}{2}=0.995,\quad\text{and } t_{n-2,1-\frac{\alpha}{2}}=t_{28,0.995}=2.763.\\ &\text{(Use ``upper tail probability'' $p=\tfrac{\alpha}{2}=0.005$, or ``probability'' $C=1-\alpha=0.99$.)}\\ \\ &\text{Reject if (the estimate of $\beta_1$) }\hat{b_1}\notin\left[-t_{28,0.995}\cdot\frac{S}{\sqrt{S_{xx}}},~~t_{28,0.995}\cdot\frac{S}{\sqrt{S_{xx}}}\right]\\ &\text{i.e. }|\hat{b_1}|>\left|t_{28,0.995}\cdot\frac{S}{\sqrt{S_{xx}}}\right|.\\ &\text{Given the test statistic }t_0=\sqrt{S_{xx}}\frac{\hat{b_1}}{S}=-7.80,\text{ the rejection criterion becomes}\\ &\quad|\hat{b_1}|>\left|t_{28,0.995}\cdot\frac{\hat{b_1}}{t_0}\right|. \qquad\text{i.e. }|t_0|>t_{28,0.995}.\\ &\text{From the data $7.8>2.763$, so $H_0$ is rejected.}\\ \end{align*} \end{document}