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December 03, 2025 02:09
ANOVA
hypothesis testing
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t-test
z-test
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A comprehensive guide to hypothesis testing, covering z-tests, t-tests, ANOVA, correlation, and regression. Useful for quick reference and exam preparation.

Hypothesis Testing Fundamentals

Basic Concepts

Hypothesis: A statement about a population parameter.

Null Hypothesis (H₀): A statement of no effect or no difference; it is assumed to be true until evidence indicates otherwise.

Alternative Hypothesis (H₁ or Hₐ): A statement that contradicts the null hypothesis; it represents what we are trying to find evidence for.

Types of Tests:

  • One-Tailed (Right-Tailed): H₁: μ > value
  • One-Tailed (Left-Tailed): H₁: μ < value
  • Two-Tailed: H₁: μ ≠ value

Type I Error (α): Rejecting H₀ when it is actually true (False Positive).

Type II Error (β): Failing to reject H₀ when it is actually false (False Negative).

Significance Level (α): The probability of making a Type I error.

Power (1-β): The probability of correctly rejecting H₀ when it is false.

Steps in Hypothesis Testing

  1. State the Hypotheses: Define H₀ and H₁.
  2. Determine the Test Statistic: Choose the appropriate test statistic (z, t, F, etc.).
  3. Set the Significance Level: Determine α (e.g., 0.05, 0.01).
  4. Calculate the Test Statistic: Compute the value of the test statistic from the sample data.
  5. Determine the p-value or Critical Value:
    • p-value: The probability of observing a test statistic as extreme as, or more extreme than, the one computed if H₀ is true.
    • Critical Value: The value(s) that define the rejection region.
  6. Make a Decision:
    • p-value Method: If p-value ≤ α, reject H₀.
    • Critical Value Method: If the test statistic falls in the rejection region, reject H₀.
  7. State the Conclusion: Interpret the decision in the context of the problem.

Z-Test vs T-Test

Z-Test

  • Population standard deviation (σ) is known.
  • Sample size is large (n ≥ 30).
  • Used for testing hypotheses about a single mean or comparing two means.

Test Statistic:

z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}

T-Test

  • Population standard deviation (σ) is unknown.
  • Sample size is small (n < 30).
  • Used for testing hypotheses about a single mean or comparing two means.

Test Statistic:

t = \frac{\bar{x} - \mu}{\frac{s}{\sqrt{n}}}

Two-Sample Tests

Two-Sample Z-Test for Means

Purpose: To test if there is a significant difference between the means of two independent populations when the population standard deviations are known or sample sizes are large.

Hypotheses:

  • H₀: μ₁ = μ₂
  • H₁: μ₁ ≠ μ₂ (two-tailed)
  • H₁: μ₁ > μ₂ (right-tailed)
  • H₁: μ₁ < μ₂ (left-tailed)

Test Statistic:
z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}

Where:

  • \bar{x}_1, \bar{x}_2 are the sample means.
  • \mu_1, \mu_2 are the population means.
  • \sigma_1, \sigma_2 are the population standard deviations.
  • n_1, n_2 are the sample sizes.

Example: Testing the difference in weight loss between a low-carb diet (n₁ = 80, x̄₁ = 13.5 lbs, s₁ = 6.59 lbs) and a low-fat diet (n₂ = 76, x̄₂ = 15.1 lbs, s₂ = 6.38 lbs) at α = 0.01.

  • H₀: μ₁ = μ₂
  • H₁: μ₁ ≠ μ₂

z = \frac{(13.5 - 15.1) - 0}{\sqrt{\frac{6.59^2}{80} + \frac{6.38^2}{76}}} ≈ -1.54

Since -2.58 < -1.54 < 2.58, fail to reject H₀. There is not sufficient evidence to conclude that the mean weight loss differs.

Paired Sample T-Test

Purpose: To test if there is a significant difference between two related populations (e.g., before and after treatment).

Hypotheses:

  • H₀: μd = 0
  • H₁: μd ≠ 0 (two-tailed)
  • H₁: μd > 0 (right-tailed)
  • H₁: μd < 0 (left-tailed)

Test Statistic:
t = \frac{\bar{d} - \mu_d}{\frac{s_d}{\sqrt{n}}}

Where:

  • \bar{d} is the mean of the differences.
  • \mu_d is the hypothesized mean difference (usually 0).
  • s_d is the standard deviation of the differences.
  • n is the number of pairs.

Example: Testing if runners ran faster after eating spaghetti (n = 6, d̄ = 3.0, sd = 3.742) at α = 0.05.

  • H₀: μd = 0
  • H₁: μd > 0

t = \frac{3.0 - 0}{\frac{3.742}{\sqrt{6}}} ≈ 1.963

Since 1.963 < 2.015, fail to reject H₀. There is not sufficient evidence to support the claim that runners run faster after eating spaghetti.

Correlation and Regression

Correlation

Purpose: To measure the strength and direction of the linear relationship between two variables.

Pearson Correlation Coefficient (r):
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n(\sum x^2) - (\sum x)^2][n(\sum y^2) - (\sum y)^2]}}

Interpretation:

  • -1 ≤ r ≤ 1
  • r > 0: Positive correlation
  • r < 0: Negative correlation
  • r ≈ 0: No correlation
  • |r| close to 1: Strong correlation

Example: Calculating correlation between non-member price (x) and member price (y) for n = 8 pairs.

r ≈ 0.857 indicates a strong positive correlation. As non-member price increases, member price tends to increase as well.

Linear Regression

Purpose: To model the relationship between two variables and predict the value of one variable based on the other.

Regression Line Equation:
\hat{y} = a + bx

Where:

  • \hat{y} is the predicted value of the dependent variable.
  • x is the independent variable.
  • a is the y-intercept.
  • b is the slope.

Formulas for Slope (b) and Y-Intercept (a):
b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}

a = \bar{y} - b\bar{x}

Example: Predicting number of mistakes (y) based on hours without sleep (x).

\hat{y} = -2.769 + 0.288x

For x = 40 hours:
\hat{y} = -2.769 + 0.288(40) ≈ 8.751 ≈ 9 mistakes.

Scatter Plots

Visual Representation: A scatter plot is a graph that displays the relationship between two variables.

Types of Relationships:

  • Positive Linear: As x increases, y increases.
  • Negative Linear: As x increases, y decreases.
  • No Relationship: Points are scattered randomly.
  • Nonlinear: Points follow a curved pattern.

Example: Plotting age (x) vs. income (y) data points on a scatter plot can reveal a positive linear relationship, indicating that income tends to increase with age.

Analysis of Variance (ANOVA)

ANOVA Fundamentals

Purpose: To test if there is a significant difference between the means of three or more independent groups.

Hypotheses:

  • H₀: μ₁ = μ₂ = μ₃ = … = μk (all means are equal)
  • H₁: At least one mean is different

Test Statistic: F = MSbetween / MSwithin

Where:

  • MSbetween is the mean square between groups.
  • MSwithin is the mean square within groups.

Degrees of Freedom:

  • dfNumerator = k - 1 (k is the number of groups)
  • dfDenominator = N - k (N is the total number of observations)

F-Distribution:

  • Always right-tailed.
  • Critical value is found using α, dfNumerator, and dfDenominator.

Sums of Squares:

  • SSTotal = ∑x² - (∑x)² / N
  • SSBetween = ∑ni (x̄i - X̄GM)²
  • SSWithin = SSTotal - SSBetween

ANOVA Calculation Steps

  1. Calculate the Group Sums, Means, and Total Mean.
  2. Calculate the Sum of Squares Total (SSTotal).
  3. Calculate the Sum of Squares Between (SSBetween).
  4. Calculate the Sum of Squares Within (SSWithin).
  5. Calculate the Mean Squares:
    • MSBetween = SSBetween / dfNumerator
    • MSWithin = SSWithin / dfDenominator
  6. Calculate the F statistic: F = MSBetween / MSWithin
  7. Find the Critical Value: Using α, dfNumerator, and dfDenominator.
  8. Make a Decision: If F > Fcrit, reject H₀.
  9. State the Conclusion.

Example

Example: Testing if average number of students in an English course differs by time of day (Morning, Afternoon, Evening) at α = 0.05.

F ≈ 4.776, Fcrit = 3.89. Since 4.776 > 3.89, reject H₀. There is sufficient evidence to conclude that mean attendance differs by time of day.