The stock market Some people think that the behavior of the...

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 $$ \begin{array}{c}{\overline{x}=1.75 \% \quad s_{x}=5.36 \% \quad \overline{y}=9.07 \%} \\ {s_{y}=15.35 \% \quad r=0.596}\end{array} $$ (a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work. (b) The mean change in January is $\overline{x}=1.75 \%$ . Use your regression line to predict the change in the index in a year in which the index rises 1.75$\%$ in January. Why could you have given this result (up to roundoff error) without doing the calculation?

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
$$
\begin{array}{c}{\overline{x}=1.75 \% \quad s_{x}=5.36 \% \quad \overline{y}=9.07 \%} \\ {s_{y}=15.35 \% \quad r=0.596}\end{array}
$$
(a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work.
(b) The mean change in January is $\overline{x}=1.75 \%$ . Use your regression line to predict the change in the index in a year in which the index rises 1.75$\%$ in January. Why could you have given this result (up to roundoff error) without doing the calculation?

Key Concepts

Prediction at the Mean

When using the regression line for prediction, if the value of the explanatory variable equals its mean, the predicted value of the response variable will equal the mean of the response variable. This is because the regression line is designed to pass through the point defined by the means of the variables, a property that simplifies interpretation and prediction at the center of the data.

Slope and Intercept Calculation

Calculating the slope and intercept of the regression line involves using formulas that incorporate the means and standard deviations of the explanatory and response variables, along with the correlation coefficient. The slope represents the change in the response variable for a one-unit change in the explanatory variable, while the intercept represents the predicted value of the response variable when the explanatory variable is zero.

Simple Linear Regression

Simple linear regression is a statistical method used to model the relationship between a single explanatory variable and a response variable by fitting a linear equation to observed data. It assumes that this relationship can be described by a straight line, making it possible to predict the response variable based on a given value of the explanatory variable.

Least-Squares Line

The least-squares line is the line that minimizes the sum of the squared differences between the observed values and the values predicted by the line. This method, known as the method of least squares, is used to obtain the best fitting straight line through the data points and is fundamental to linear regression analysis.

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