Statistics and Testing Concepts

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

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 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.

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}) )

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 )

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.

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.

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.