Question

The stock market Some people think that the behavior of the stock market in January predicts its behavior for the rest of the year. Take the explanatory variable x to be the percent change in a stock market index in January and the response variable y to be the change in the index for the entire year. We expect a positive correlation between x and y because the change during January contributes to the full year’s change. Calculation from data for an 18-year period gives x¯=1.75% sx=5.36% y¯=9.07% sy=15.35% r=0.596

a. Find the equation of the least-squares line for predicting full-year change from January change.

b. Suppose that the percent change in a particular January was 2 standard deviations above average. Predict the percent change for the entire year without using the least-squares line.

Short answer

Part (a)$\stackrel{⏜}{y}=6.0831+1.7068x$

Part (b) The predicted height is$9.07\%$

Step-by-step solution

Part (a) Step 1: Given information and Step 2: Explanation 

$$\begin{gathered} r=0.596 \\ \overline{x}=1.75 \\ {s}_x=5.36 \\ \overline{y}=9.07 \\ {s}_y=15.35 \end{gathered}$$

The general equation of the regression line is:

$\stackrel{⏜}{y}=a+bx$

The slope of the regression line is:

$$\begin{gathered} b=r\frac{s_y}{s_x} \\ =0.596\times \frac{15.35}{5.36} \\ =1.7068 \end{gathered}$$

And the intercept is as:

$$\begin{gathered} a=\overline{y}-b\overline{x} \\ =9.07-1.7068(1.75) \\ =6.0831 \end{gathered}$$

Thus, the equation of the regression line is:

$$\begin{gathered} \stackrel{⏜}{y}=a+bx \\ \Rightarrow \stackrel{⏜}{y}=6.0831+1.7068x \end{gathered}$$

Part (b) Step 1: Explanation

Thus, we have,

$$\begin{gathered} \stackrel{⏜}{y}=6.0831+1.7068x \\ =6.0831+1.7068(1.75) \\ =9.07 \end{gathered}$$

Thus the predicted height is $9.07\%$