Statistics and Testing Concepts

The File Drawer Effect

• Definition: Failing to publish uninteresting results, often occurring when researchers fail to reject the null hypothesis (FTR Ho).

Central Limit Theorem (CLT)

• Statement of CLT: With a sample size of 64 drawn from a distribution with mean (µ) = 50 and standard deviation (σ) = 12, the sample mean (X) is approximately normally distributed with:
  • Mean: 50
  • Standard Deviation: ( \sigma_{X} = \frac{12}{\sqrt{64}} = 1.5 )
• Z-Score Calculation: For a score of 48, the standardized residual (z-score) is calculated as:
  • ( z = \frac{48 - 50}{1.5} = -1.33 )

Type I and Type II Errors

• Type I Error (α): Rejecting the null hypothesis (Ho) when it is true.
  • Probability of Type I error is denoted as P(Type I error) = α.
• Type II Error (β): Failing to reject the null hypothesis when it is false.
  • Probability of Type II error is denoted as P(Type II error) = β.
• Power of a Test: The probability of correctly rejecting a false null hypothesis, denoted as 1 - β.

Benford's Law

• Use: To test the legitimacy of data by examining the distribution of the first digit (logarithmic pattern).
• Formula: ( \text{P(first digit} = d) = \log_{10}(1 + \frac{1}{d}) )

Chi-Square Tests Example 1: Commuter Status vs. Yankee Fans

• Data Summary:
  • Commuters: 20 fans, 25 non-fans (Total: 45)
  • Non-Commuters: 10 fans, 50 non-fans (Total: 60)
• Hypotheses:
  • Ho: Commuter status is independent of being a Yankee fan.
  • Ha: Commuter status is not independent of being a Yankee fan.
• Chi-Square Result:
  • Test Statistic = 9.72, Degrees of Freedom (DF) = 1, P-Value = 0.0018
  • Since P-Value < α (0.05), reject Ho.
• Conclusion: Evidence suggests a relationship between commuter status and being a Yankee fan.
• Odds Ratio Calculation:
  • ( \frac{20 \cdot 50}{10 \cdot 25} = 4 )
  • Interpretation: Odds of a commuter being a Yankee fan is 4 times that of a non-commuter.
• Expected Count for Commuters who are Non-Yankee Fans:
  • ( \frac{45 \cdot 75}{105} = 32.14 )

Chi-Square Tests Example 2: Party Affiliation and Candidate Support

• Data Summary:
  • Democrats, Republicans, Independents supporting 4 candidates.
• Hypotheses:
  • Ho: No relationship between party affiliation and candidate supported.
  • Ha: Relationship exists.
• Chi-Square Result:
  • Test Statistic = 37.73, P-Value < 0.0001
  • Reject Ho, indicating relationship exists.

Chi-Square Tests Example 3: Car Size and Shirt Size

• Data Summary:
  • Relationship studied between car size categories and shirt sizes (Small, Medium, Large, X-Large).
• Hypotheses:
  • Ho: Shirt size and car size are independent.
  • Ha: They are not independent.
• Chi-Square Result:
  • Test Statistic = 360.76, P-Value < 0.0001
  • Reject Ho, indicating a relationship.
• Gamma Correlation Coefficient:
  • ( \text{Gamma} = 0.54953 ) indicates a strong positive relationship, suggesting larger cars are associated with larger shirt sizes.

Chi-Square Tests Example 4: Favorite Crayon Color Distribution

• Data Summary:
  • Frequency of respondents' favorite colors.
• Hypotheses:
  • Ho: Favorite colors are uniformly distributed.
  • Ha: They are not.
• Chi-Square Result:
  • Test Statistic = 23.72, P-Value = 0.0013
  • Reject Ho, indicating favorite colors are not uniformly distributed.

Shapiro-Wilk Test

• Understand how to perform this test for normality assessment of the dataset.