Bull Markets A bull market is defined as a market condition in which the price of a security rises for an extended period of time. A bull market in the stock market is often defined as a condition in which a market rises by 20% or more without a 20% decline. The data to the right represent the number of months and percentage change in the S&P 500 (a group of 500 stocks) during the 25 bull markets dating back to 1929 (the year of the famous market crash).


b. Determine the linear correlation coefficient between months and percent change.

Verified step by step guidance

1. Step 1: Organize the data into two lists or columns: one for 'Bull Months' and one for 'Percent Change'. Ensure all 25 pairs of data points are included.
2. Step 2: Calculate the mean (average) of the 'Bull Months' data and the mean of the 'Percent Change' data. Use 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: \[ (x_i - \bar{x}) \quad \text{and} \quad (y_i - \bar{y}) \] for each \(i\) from 1 to 25.
4. Step 4: Calculate the covariance between 'Bull Months' and 'Percent Change' using: \[ \text{Cov}(x,y) = \frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) \] Also calculate the standard deviations of both variables: \[ s_x = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (x_i - \bar{x})^2} \] \[ s_y = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (y_i - \bar{y})^2} \]
5. Step 5: Finally, compute the linear correlation coefficient \(r\) using the formula: \[ r = \frac{\text{Cov}(x,y)}{s_x s_y} \] This value will indicate the strength and direction of the linear relationship between the number of months and percent change during bull markets.

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 indicate a strong positive correlation, values close to -1 indicate a strong negative correlation, and values near 0 suggest no linear correlation.

Scatterplot and Data Visualization

A scatterplot is a graphical representation that displays the relationship between two variables by plotting data points on a Cartesian plane. Visualizing data helps identify patterns, trends, or potential correlations, which is essential before calculating the correlation coefficient to understand the nature of the relationship.

Interpretation of Market Data in Statistics

Understanding market data like bull months and percent change involves applying statistical methods to real-world financial contexts. Recognizing how time duration (months) relates to percent change in stock prices helps in interpreting trends and making informed decisions based on statistical analysis.