The Wall Street Journal's Shareholder Scoreboard Analysis
The Chi-Squared Test for Shareholder Scoreboard Performance Analysis
The Wall Street Journal's Shareholder Scoreboard tracks the performance of 1,000 major U.S. companies. The performance of each company is rated based on the annual total return, including stock price changes and the reinvestment of dividends.
The chi-squared test indicates a significant difference in performance between the largest companies and the entire 1000 companies in the Shareholder Scoreboard (p < 0.05) including stock price the Shareholder Scoreboard, we can perform a hypothesis test using a chi-squared goodness-of-fit test.
The null hypothesis (H0) is that the distribution of ratings for the largest companies is the same as the distribution of ratings for the entire 1000 companies.
The alternative hypothesis (Ha) is that the distribution of ratings for the largest companies is different from the distribution of ratings for the entire 1000 companies including stock price.
Given the observed frequencies for each rating category:
A (Top 20%): 5
B (Next 20%): 8
C (Middle 20%): 15
D (Next 20%): 20
E (Bottom 20%): 12
And the expected frequencies for each rating category (assuming an equal distribution):
A: (1/5) * 60 = 12
B: (1/5) * 60 = 12
C: (1/5) * 60 = 12
D: (1/5) * 60 = 12
E: (1/5) * 60 = 12
Let's calculate the chi-squared test statistic and the p-value using the formula:
χ² = Σ((Observed - Expected)² / Expected)
1. Calculate the test statistic:
χ² = ((5 - 12)² / 12) + ((8 - 12)² / 12) + ((15 - 12)² / 12) + ((20 - 12)² / 12) + ((12 - 12)² / 12)
χ² = (49/12) + (4/3) + (9/12) + (64/12) + 0
χ² = 4.08333 + 1.33333 + 0.75 + 5.33333 + 0
χ² = 11.5
2. Calculate the degrees of freedom (df):
df = number of categories - 1
df = 5 - 1
df = 4
3. Use a chi-squared distribution table or calculator to find the p-value associated with χ² = 11.5 and df = 4.
Assuming you find that the p-value is less than the significance level of 0.05, you would reject the null hypothesis and conclude that the largest companies' performance differs from the performance of the 1000 companies in the Shareholder Scoreboard.
The Wall Street Journal's Shareholder Scoreboard tracks the performance of 1000 major U.S. companies (The Wall Street Journal, March 10, 2003). The performance of each company is rated based on the annual total return, including stock price changes and the reinvestment of dividends. Ratings are assigned by dividing all 1000 companies into five groups from A (top 20%), B (next 20%), to E (bottom 20%). Shown here are the one-year ratings for a sample of 60 of the largest companies. Do the largest companies differ in performance from the performance of the 1000 companies in the Shareholder Scoreboard? Use α = .05. A=5, B=8, C=15, D=20, E=12 1. What is the test statistic? 2. What is the p-value? 1. The test statistic is χ² = 11.5 2. The p-value associated with χ² = 11.5 and df = 4 is less than the significance level of 0.05, indicating that the largest companies differ in performance from the entire 1000 companies in the Shareholder Scoreboard.