Statistics Formulas and Hypothesis Testing Reference

Chapter 1: Margin of Error and Sample Size

• Margin of Error (ME) = (Upper bound - Lower bound) / 2
• Sample size for a proportion type: n = z² × p̂ × (1 - p̂) / ME²
• Standard Deviation = √(pq/n)
• Width = (√n / √new n) × Old width
• Confidence Interval: Helps make correct business decisions by obtaining more information about the population parameter.

Chapter 2: Mean Intervals and T-Tests

• True mean interval for one sample: Use 8:TInterval.
• Critical value of t^*: VARS → InvT(area, df), where df = n - 1 and area = ((1 - confidence level) / 2) + confidence level.
• Sample size for T-test: n = (z × σ / ME)²
• Standard Error of the Mean: SE = s / √n. Alternatively, use 1:T-test to find Sx / √n.
• A t-interval has a larger Margin of Error than a z-interval with a similarly chosen confidence level.

Chapter 3: Proportion Tests and P-Values

• P-value for a proportion test: Use 5: 1-propZtest. Use >p for "more than" or "higher than" and <p for "less than" or "decreased."
• Critical z^* for a two-tailed H_a: 1) 1 - (significance level / 2); 2) VARS → 3:InvNorm(result, 0, 1).
• Use a T-test for quantitative data when testing for a mean and σ is unknown.
• If P-value > α (0.05), fail to reject H₀. If P-value < α, reject H₀.
• A higher Standard Error (SE) results in a higher P-value and a lower Z-test statistic.
• Finding a probability: 1) SD = √(pq/n); 2) 2:normalcdf(lower, upper, p̂, SD). Use 999 for the upper bound when finding "more than."

Chapter 5: Statistical Power and Error Types

• Power = 1 - β. If α increases, β decreases.
• Type I Error: Rejecting H₀ when it is true.
• Type II Error: Failing to reject H₀ when it is false.
• Confidence Interval for true mean length: Use 8:TInterval.
• Determining if Error I or Error II occurred: Use 2:T-test.

Chapter 6: Comparing Means and Paired Data

• Hypothesis test: 1) Identify H₀ and H_a; 2) Use 2-SampT-Test.
• Standard Error of the mean difference in paired data: 1) Enter differences in L1; 2) Calc → 1-Var Stats; 3) SE = Sx / √n.
• Standard Error for the difference between two means: √((sd₁² / n₁) + (sd₂² / n₂)).
• Interpretation of Confidence Interval (CI): If 0 falls inside the CI, you cannot conclude there is a significant difference.
• Calculating Confidence Interval: Use 0: 2-SampTInt.

Chapter 7: Chi-Square Analysis

• Determining the test statistic: 2nd Matrix → Edit [A] → Stat → X²-test.
• Finding the P-value with X² and degrees of freedom: DISTR → X²CDF(X², 999, df).
• Test statistic with one row (Goodness of Fit): 1) ∑ of values / number of columns = expected values; 2) L1 (Observed), L2 (Expected); 3) X²GOF-Test (df = columns - 1).

Chapter 8: Regression Confidence Intervals

• Confidence Interval for slope: b₁ ± t^* × SE(b₁).

Chapter 9: Regression Diagnostics

• Leverage Point: An observation that is far from the mean of X.
• Influential Point: A point that significantly changes the slope of the regression line.
• Residual: An observation that is vertically far away from the main cluster.