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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 $$\begin{array}{c}\bar{x}=1.75 \% \quad s_{x}=5.36 \% \quad \bar{y}=9.07 \% \\\ s_{y}=15.35 \% \quad r=0.596\end{array}$$ (a) Find the equation of the least-squares line for predicting full-year change from January change. Show your work. (b) 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. Show your work.

Step by step solution

01

Understand the Given Data

We need to find the equation of the least-squares regression line for predicting the full-year stock market index change from the January change. The given data are: \( \bar{x} = 1.75\% \), \( s_x = 5.36\% \), \( \bar{y} = 9.07\% \), \( s_y = 15.35\% \), and the correlation coefficient \( r = 0.596 \).

02

Calculate the Slope of the Regression Line

The slope \( b \) of the least-squares regression line is given by \( b = r \frac{s_y}{s_x} \). Substituting the given values, we have:\[ b = 0.596 \frac{15.35}{5.36} = 1.706 \].

03

Calculate the Intercept of the Regression Line

The intercept \( a \) of the regression line is found using \( a = \bar{y} - b\bar{x} \). Substituting into the formula:\[ a = 9.07 - 1.706 \times 1.75 = 6.0835 \].

04

Write the Equation of the Regression Line

The least-squares regression line is given by \( y = a + bx \). Substituting the calculated values, the equation is:\[ y = 6.0835 + 1.706x \].

05

Determine the Effect of a January Change

If a January change is 2 standard deviations above average, we calculate it as \( x = \bar{x} + 2s_x \). This is:\[ x = 1.75 + 2 \times 5.36 = 12.47\% \].

06

Calculate the Expected Year-Long Change Using Regression

Using the regression line equation \( y = 6.0835 + 1.706x \), substitute \( x = 12.47\% \):\[ y = 6.0835 + 1.706 \times 12.47 = 27.32\% \].

07

Predict Without Using the Least Squares Line

To predict without the regression line, use the concept of standard deviation units and correlation. Since January's percent change is 2 standard deviations above the mean, and knowing \( r = 0.596 \), the prediction for the year is:\[ \bar{y} + 2r \cdot s_y = 9.07 + 2 \times 0.596 \times 15.35 = 27.37\% \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Correlation Coefficient

The correlation coefficient, often denoted as \( r \), is a statistical measure that describes the strength and direction of a linear relationship between two variables. In our stock market example, \( r = 0.596 \). This value signifies a positive correlation, meaning that as the January stock index increases, the annual stock market index is also likely to increase. This occurs because the positive value suggests that both variables tend to move in the same direction.Several key points about the correlation coefficient:

• It ranges from -1 to 1.
• A value close to 1 indicates a strong positive correlation.
• A value close to -1 indicates a strong negative correlation.
• A value around 0 indicates little to no linear correlation.

While helpful, it's crucial to remember that correlation doesn't imply causation. So, a high correlation doesn't mean that January's performance causes the annual performance to increase.

Explanatory Variable and Response Variable

In a study of relationships between variables, it's important to know which one is the explanatory variable and which is the response variable. These roles inform how we set up and interpret an analysis.For our exercise, we have:

• The explanatory variable, denoted \( x \), is the percent change in the stock market index in January. It's considered explanatory because we are investigating its potential impact on the yearly change.
• The response variable, denoted \( y \), is the percent change in the stock index for the entire year. This is what we predict based on the explanatory variable.

Understanding these definitions helps in constructing models. The relationship modeled by the regression line assumes that changes in \( x \) can explain or predict changes in \( y \). However, remember, model predictions are only as good as the data assumptions and may not cover all factors affecting the response variable.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In our exercise, we've seen that \( s_x = 5.36\% \) for January's stock change and \( s_y = 15.35\% \) for the year's change.To understand standard deviation:

• A smaller standard deviation means the data points tend to be close to the mean.
• A larger standard deviation means there is more spread among the data points.

It's also applied in predicting future values in the context of stock market performance. For example, a change in January that's 2 standard deviations above average was used as \( x = \bar{x} + 2s_x = 12.47\% \). This figure helps forecast potential outcomes using the variability inherent in \( x \). This approach helps gauge how unusual a particular value is compared to the historical performance.