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The Dow Jones Industrial Average (DJIA) and the Standard \& Poor's 500 Index (S\&P 500) are both used to measure the performance of the stock market. The DJIA is based on the price of stocks for 30 large companies; the \(\mathrm{S\&P} 500\) is based on the price of stocks for 500 companies. If both the DJIA and S\&P 500 measure the performance of the stock market, how are they correlated? The following data show the daily percent increase or daily percent decrease in the DJIA and S\&P 500 for a sample of nine days over a three-month period (The Wall Street Journal, January 15 to March 10, 2006). \\[\begin{array}{lllllllll}\text { DJIA } & .20 & .82 & -.99 & .04 & -.24 & 1.01 & .30 & .55 & -.25 \\ \text { S\&P 500 } & .24 & .19 & -.91 & .08 & -.33 & .87 & .36 & .83 & -.16\end{array}\\] a. Show a scatter diagram.b. Compute the sample correlation coefficient for these data. c. Discuss the association between the DJIA and S\&P \(500 .\) Do you need to check both before having a general idea about the daily stock market performance?

Step by step solution

01

Visualize the Data

To create a scatter plot, plot each paired observation on a two-dimensional graph, where the x-axis represents DJIA changes and the y-axis represents S&P 500 changes. Each pair will be one point on the graph.

02

Calculate Means of Data Sets

First, calculate the mean (average) for both DJIA and S&P 500 data. For DJIA, add all data points and divide by the number of data points (9). Repeat the same for S&P 500. Mean for DJIA: \(\bar{x} = \frac{2.44}{9}\), Mean for S\&P 500: \(\bar{y} = \frac{2.07}{9}\).

03

Compute the Covariance

Calculate the covariance between DJIA and S&P 500 data. Use the formula \(\text{Cov}(X,Y) = \frac{1}{n-1}\sum (x_i - \bar{x})(y_i - \bar{y})\). Substitute each value, subtract the mean, and sum their products. Divide by 8 (\(n-1\)) for the covariance.

04

Compute the Standard Deviations

Find the standard deviation for both data sets. Use \(s_x = \sqrt{\frac{1}{n-1}\sum (x_i - \bar{x})^2}\) for DJIA and similarly for S&P 500. Each x_i and y_i represents the data point, and subtract the mean, square the result, sum up, divide by \(n-1\), and take the square root.

05

Calculate the Correlation Coefficient

Use the covariance and respective standard deviations to find the sample correlation coefficient \(r\). This uses the formula \(r = \frac{\text{Cov}(X,Y)}{s_x s_y}\). Substitute the calculated covariance and standard deviations from previous steps to get \(r\).

06

Interpret the Results

Consider the correlation coefficient \(r\). If \(r\) is close to 1, there is a strong positive association; if close to -1, a strong negative association; if close to 0, little to no correlation exists. Discuss this association in terms of DJIA and S&P 500's ability to capture the daily stock market performance.

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Key Concepts

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

Scatter Plot

A scatter plot is a type of graph used to visualize the relationship between two sets of data. By plotting each pair of values on a graph where one variable is represented on the x-axis and another on the y-axis, you can easily see how each set behaves in relation to the other.
For the stock markets, such as the DJIA and S&P 500, a scatter plot can help you visually assess their correlation. Place the daily percent changes of the DJIA on the x-axis and those of the S&P 500 on the y-axis. Each day’s data point forms a coordinate on the plot.
This visual representation is crucial because it offers an immediate, intuitive sense of how closely tied the two indices are. If the points are closely clustered around an upward sloping line, it suggests a positive correlation, meaning both indices tend to move in the same direction. Conversely, if they are scattered randomly across the plot, it might indicate no substantial correlation.

Covariance

Covariance is a statistical measure used to determine the directional relationship between two variables. When you calculate the covariance between the DJIA and S&P 500, you are examining whether the changes in one index are in sync with the changes in the other.
To calculate covariance, you subtract the mean of each data set from individual data points, multiply the results for each pair, and then average those products. In formula terms, it's represented as \[ \text{Cov}(X, Y) = \frac{1}{n-1}{\sum (x_i - \bar{x})(y_i - \bar{y})} \], where \( n \) is the number of paired observations, and \( \bar{x} \) and \( \bar{y} \) are the means.
If the covariance is positive, both the DJIA and S&P 500 tend to increase or decrease together. If negative, one index's increase corresponds with the other's decrease. A covariance near zero suggests no linear relationship between the indices. Understanding covariance helps in evaluating how different stock indices' movements are connected, aiding in financial decisions and investment strategies.

Standard Deviation

Standard deviation measures the amount of variation or dispersion in a set of values, allowing us to understand how spread out the data is from the average. When applied to stock indices like the DJIA and S&P 500, standard deviation tells us how much each index fluctuates day-to-day.
Calculating standard deviation involves taking each data point's difference from the mean, squaring it, finding the average of those squared differences, and finally taking the square root of that average. The formula is \[ s_x = \sqrt{\frac{1}{n-1}{\sum (x_i - \bar{x})^2}} \] for each data set.
A high standard deviation indicates that the index experiences more volatility, with values spread out over a wide range. A low standard deviation suggests less volatility, with values closely clustered around the mean. In the context of stock market performance, understanding standard deviation helps investors assess risk, as more volatile investments may yield higher returns but also carry greater risk of loss.

Stock Market Performance

Stock market performance is often gauged by key indices like the DJIA and S&P 500, representing different aspects of economic activity. These indices provide insights into the general direction and health of the market.
The DJIA tracks 30 large publicly traded companies, while the S&P 500 covers 500, offering a broader market view. To understand their performance correlation, both are analyzed together using a correlation coefficient. This metric illustrates how strongly indices are associated.
If the correlation coefficient is close to 1, it shows a strong positive correlation, suggesting that both indices often move in tandem. A coefficient near -1 indicates an inverse relationship, while a value near zero implies little to no overall correlation. Assessing these correlations helps determine whether one needs to check both indices to get a comprehensive view of the stock market's health, ultimately aiding in making informed investment decisions.