Statistical Inference and Hypothesis Testing Procedures

Central Limit Theorem (CLT)

Check the sample size n. If n is 30 or more, the sampling distribution is normal. The mean of the sample mean equals the population mean. The standard error is calculated as σ divided by the square root of n.

Z Distribution for Population Mean

• Identify values: Sample mean, population mean, σ, and n.
• Compute standard error: SE = σ / √n.
• Compute z: z = (sample mean - population mean) / standard error.
• Use the z-table to find the probability or critical value.
• Confidence interval: Sample mean ± (z-critical × standard error).

T Distribution for Population Mean

• Find the sample mean and sample standard deviation (s).
• Standard error: SE = s / √n.
• Determine the degrees of freedom (df).
• Compute t: t = (sample mean - population mean) / standard error.
• Use the t-table with the appropriate df.
• Confidence interval: Sample mean ± (t-critical × standard error).

Hypothesis Testing for One Population

1. Write hypotheses: State the null hypothesis (H₀) and the alternative hypothesis (H₁).
2. Choose test: Select between a z-test or t-test.
3. Calculate test statistic: Compute the relevant value.
4. Find p-value or critical value: Determine the significance.
5. Decision:
   • If the p-value ≤ α → Reject H₀.
   • If the p-value > α → Fail to reject H₀.

Two Means for Independent Samples

• Get values: mean₁, mean₂, s₁, s₂, n₁, n₂.
• Standard error: SE = √((s₁² / n₁) + (s₂² / n₂)).
• Compute t: t = (mean₁ - mean₂) / standard error.
• Find DF: Determine degrees of freedom.
• Compare the result with the table or p-value.

Two Means for Dependent Samples

Use when: Analyzing the same group before and after an event.

• Find the differences (d = value₁ - value₂).
• Find the mean of the differences and the standard deviation of the differences (s_d).
• Standard error: SE = s_d / √n.
• Compute t: t = mean difference / standard error.
• Degrees of freedom: df = n - 1.

Chi-Square Test of Independence

• State H₀: Observed values equal expected values.
• Compute expected: E = total × proportion.
• Compute Chi-Square (χ²): ∑((observed - expected)² / expected).
• Get DF: Determine degrees of freedom.
• Compare the result with the table or p-value.

One-Way ANOVA for Multiple Groups

Use when: Comparing three or more groups.

• H₀: All means are equal.
• H₁: At least one mean is different.
• Use Excel ANOVA: Single Factor for calculations.
• Look at the p-value.
• If p ≤ α → Reject H₀.