[DATA] Bear Markets A bear market in the stock market is defined as a condition in which the market declines by 20% or more over the course of at least two months. The following data represent the number of months and percentage change in the S&P 500 (a group of 500 stocks) for a sample of bear markets.
b. Determine the linear correlation coefficient between months and percent change.


c. Does a linear relation exist between duration of the bear market and market performance?

Verified step by step guidance

1. Step 1: Organize the data into two lists or columns: one for the 'Months' and one for the 'Percent Change'. This will help in calculating the necessary statistics for the correlation coefficient.
2. Step 2: Calculate the mean (average) of the 'Months' data and the mean of the 'Percent Change' data using the formulas: \[ \bar{x} = \frac{1}{n} \sum_{i=1}^n x_i \] \[ \bar{y} = \frac{1}{n} \sum_{i=1}^n y_i \] where \(x_i\) are the months and \(y_i\) are the percent changes.
3. Step 3: Compute the deviations from the mean for each data point: \[ (x_i - \bar{x}) \quad \text{and} \quad (y_i - \bar{y}) \] Then calculate the products of these deviations for each pair: \[ (x_i - \bar{x})(y_i - \bar{y}) \]
4. Step 4: Calculate the sums needed for the correlation coefficient formula: \[ S_{xy} = \sum (x_i - \bar{x})(y_i - \bar{y}) \] \[ S_{xx} = \sum (x_i - \bar{x})^2 \] \[ S_{yy} = \sum (y_i - \bar{y})^2 \]
5. Step 5: Use the formula for the Pearson correlation coefficient \(r\): \[ r = \frac{S_{xy}}{\sqrt{S_{xx} \times S_{yy}}} \] This value will indicate the strength and direction of the linear relationship between the duration of the bear market and the percent change in the market.

Key Concepts

Linear Correlation Coefficient

The linear correlation coefficient, often denoted as r, measures the strength and direction of a linear relationship between two quantitative variables. It ranges from -1 to 1, where values close to 1 or -1 indicate strong positive or negative linear relationships, respectively, and values near 0 suggest no linear correlation.

Scatterplot and Linear Relationship

A scatterplot visually displays the relationship between two variables by plotting data points on a coordinate plane. Observing the pattern of points helps determine if a linear relationship exists, which is essential before calculating correlation or fitting a regression line.

Interpretation of Correlation in Context

Interpreting the correlation coefficient requires understanding the context of the data. In this case, it involves assessing whether the duration of bear markets (months) is associated with the magnitude of percent change in the S&P 500, helping to infer if longer bear markets correspond to larger declines.