91Ó°ÊÓ

In the least-squares line \(\hat{y}=5-2 x,\) what is the value of the slope? When \(x\) changes by 1 unit, by how much does \(\hat{y}\) change?

What is the symbol used for the population correlation coefficient?

What is the symbol used for the slope of the population least-squares line?

For a fixed confidence level, how does the length of the confidence interval for predicted values of \(y\) change as the corresponding \(x\) values become further away from \(\bar{x} ?\)

An Internet advertising agency is studying the number of "hits" on a certain web site during an advertising campaign. It is hoped that as the campaign progresses, the number of hits on the web site will also increase in a predictable way from one day to the next. For 10 days of the campaign, the number of hits \(\times 10^{5}\) is shown: $$\begin{array}{l|rrrrrrrrrr} \hline \text { Day } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\\\\hline \text { Hits } \times 10^{5} & 1.2 & 3.5 & 4.4 & 7.2 & 6.9 & 8.3 & 9.0 & 11.2 & 13.1 & 14.6 \\\\\hline\end{array}$$ (a) To construct a serial correlation, we use data pairs \((x, y)\) where \(x=\) original data and \(y=\) original data shifted ahead by one time period. Verify that the data set \((x, y)\) for serial correlation is shown here. (For discussion of serial correlation, see Problem 15.) $$\begin{array}{c|ccccccccc}\hline x & 1.2 & 3.5 & 4.4 & 7.2 & 6.9 & 8.3 & 9.0 & 11.2 & 13.1 \\\\\hline y & 3.5 & 4.4 & 7.2 & 6.9 & 8.3 & 9.0 & 11.2 & 13.1 & 14.6 \\\\\hline\end{array}$$ (b) For the \((x, y)\) data set of part (a), compute the equation of the sample least-squares line \(\hat{y}=a+b x .\) If the number of hits was \(9.3\left(\times 10^{5}\right)\) one day, what do you predict for the number of hits the next day? (c) Compute the sample correlation coefficient \(r\) and the coefficient of determination \(r^{2} .\) Test \(\rho>0\) at the \(1 \%\) level of significance. Would you say the time series of web site hits is relatively predictable from one day to the next? Explain.