(Solved): Can someone explain why the 0.23 (Beta hat) turns negative? 15) You have collected data for the 5 ...

15) You have collected data for the 50 U.S. states and estimated the following relationship between the change in the unemployment rate from the previous year ( $\Delta \mathrm{ur})$ and the growth rate of the respective state real GDP $\left({\mathrm{g}}_{\mathrm{y}}\right)$. The results are as follows $\begin{array}{l}\widehat{\Delta \mathrm{ur}}=2.81-0.23\times g_{\mathrm{y}},\mathrm{R}2=0.36,\mathrm{SER}=0.78\\ (0.12)(0.04)\end{array}$ Assuming that the estimator has a normal distribution, the $95\%$ confidence interval for the slope is approximately the interval A) $[-0.31,0.15]$ B) $[2.57,3.05]$ C) $[-0.33,-0.13]$ D) $[-0.31,-0.15]$ Answer: D The estimated value of the coefficient $\beta$ is 0.23 , with a standard error of 0.04 . Using the standard normal approximation, the $95\%$ confidence interval for $\beta$ is required to be calculated. In the standard normal probability distribution table, the $z$-value corresponding to $0.4750(=0.95/2)$ is 1.96 ; therefore, the $95\%$ confidence interval for $\beta$ can be calculated as follows: $\begin{array}{rl}95\%\text{ confidence interval }& =\hat{\beta }\pm 1.96(\text{ standard error of }\hat{\beta })\\ & =0.23\pm 1.96(0.04)\\ & =-0.23\pm 0.08\\ & =-0.23-0.08\text{ and }-0.23+0.08\\ & =-0.31\text{ and }-0.15\end{array}$ Therefore, the $95\%$ confidence interval for $\beta$ is from -0.31 and -0.15 ; is correct.