\[ y_{i}=\beta_{1}+\beta_{2} x_{i 2}+\beta_{3} x_{i 3}+\varepsilon_{i}=x_{i}^{\prime} \beta+\varepsilon_{i} \] a. Explain how the ordinary least squares estimator for \( \beta \) is determined and derive an expression for \( b \). b. Which assumptions are needed to make \( b \) an unbiased estimator for \( \beta \) ? c. Explain how a confidence interval for \( \beta_{2} \) can be constructed. Which additional d. Explain how one can test the hypothesis that \( \beta_{3}=1 \).

A random sample of 69 produced the summary statistics \( \bar{x}=0.326 \) and \( s^{2}=0.064 \). Complete parts a and \( b \). State and interpret the results of the test. A. Reject \( H_{0} \). There is insufficient evidence to indicate that the true population mean is less than 0.37 at \( \alpha=0.10 \). Bo not reject \( H_{0} \). There is sufficient evidence to indicate that the true population mean is less than 0.37 at \( \alpha=0.10 \). C. Reject \( H_{0} \). There is sufficient evidence to indicate that the true population mean is less than 0.37 at \( \alpha=0.10 \). Do not reject \( H_{0} \). Thefe is insufficient evidence to indicate that the true population mean is less than 0.37 at \( \alpha=0.10 \).

A variable \( x \) is normally distributed with mean 22 and standard deviation 3. Round your answers to the nearest hundredth as needed. a) Determine the \( z \)-score for \( x=27 \). b) Determine the \( z \)-score for \( x=20 \). c) What value of \( x \) has a \( z \)-score of 2.33 ? d) What value of \( x \) has a \( z \)-score of -0.3 ? \( x= \) e) What value of \( x \) has a \( z \)-score of 0 ?

University researchers published a working paper. One of the objectives of the study was to determine whether ride-share drivers actually make a profit. Data collected from a sample of 1,126 ride-share drivers yielded a mean monthly profit \( \bar{x}=\$ 661 \) with standard deviation \( s=\$ 581 \). A ride-share company executive reports that \( \mu \), the true mean monthly profit of all its rideshare drivers, is \( \$ 1,000 \). Use \( \alpha=0.01 \) to test the executive's figure. Complete parts a through e below. b. Find the rejection region for the test. Select the correct choice below and fill in the answer box(es) within your choice. (Round to two decimal places as needed.) A. \( z \) < \( z>\square \) C. \( z<\square \) or \( z>\square \)

A random sample of 69 produced the summary statistics \( \bar{x}=0.326 \) and \( \mathrm{s}^{2}=0.064 \). Complete parts a and \( b \). a. Test the null hypothesis that \( \mu=0.37 \) against the alternative hypothesis that \( \mu<0.37 \), using \( \alpha=0.10 \). What is the value of the test statistic? \( z=-1.44 \) (Round to two decimal places as needed.) Find the \( p \)-value. p-value \( =\square \) (Round to three decimal places as needed.)

\( 1 \leftarrow \begin{array}{l}\text { Researchers conducted a study on using a chief executive officer's facial structure to predict a firm's financial } \\ \text { performance. The facial width-to-height ratio (WHR) for each in a sample of } 52 \text { CEOs at publicly traded firms was } \\ \text { determined. The sample resulted in } \bar{x}=1.55 \text { and } s=0.36 \text {. An analyst wants to predict the financial performance of a } \\ \text { firm based on the value of the true mean facial WHR of CEOs. The analyst wants to use the value of } \mu=2.1 \text {. Do yo } \\ \text { recommend he use this value? Conduct a test of hypothesis for } \mu \text { to help you answer the question. Specify all the } \\ \text { elements of the test, including } H_{0}, H_{a} \text {, test statistic, } p \text {-value, and your conclusion. Test at } \alpha=0.01 \text {. } \\ \text { What is the value of the test statistic? } \\ z=-11.02 \text { (Round to two decimal places as needed.) } \\ \text { Find the p-value. } \\ \text { p-value }=\square \text { (Round to three decimal places as needed.) }\end{array} \).