Introduction to Hypothesis Testing

What is Hypothesis Testing?

Hypothesis testing is a formal procedure for using sample data to evaluate claims about population parameters.

The Hypothesis Testing Framework

Step 1: State the Hypotheses

Every test has two hypotheses:

Step 2: Choose Significance Level (α)

The significance level α is the probability of rejecting H₀ when it’s actually true (Type I error).

Common α	Confidence Level
0.10	90%
0.05	95% (most common)
0.01	99%

Step 3: Collect Data and Calculate Test Statistic

The test statistic measures how far your sample result is from the null hypothesis value, in standard error units.

For a mean: z = (x̄ - μ₀) / (σ/√n) or t = (x̄ - μ₀) / (s/√n)

For a proportion: z = (p̂ - p₀) / √(p₀(1-p₀)/n)

Step 4: Find the P-Value

The p-value is the probability of getting a test statistic as extreme as (or more extreme than) the observed value, assuming H₀ is true.

P-value	Evidence against H₀
p > 0.10	Weak or none
0.05 < p ≤ 0.10	Moderate
0.01 < p ≤ 0.05	Strong
p ≤ 0.01	Very strong

Step 5: Make a Decision

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Types of Tests

One-Tailed vs Two-Tailed Tests

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Complete Example: One-Sample Z-Test

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Type I and Type II Errors

When making decisions, we can make two types of errors:

	H₀ True	H₀ False
Reject H₀	Type I Error (α)	Correct Decision (Power)
Fail to reject H₀	Correct Decision	Type II Error (β)

Factors Affecting Power

Factor	Effect on Power
↑ Sample size (n)	↑ Power
↑ Significance level (α)	↑ Power
↑ True effect size	↑ Power
↓ Population variability (σ)	↑ Power

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Critical Value Approach (Alternative)

Instead of p-values, you can use critical values:

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Statistical vs Practical Significance

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Common Mistakes in Hypothesis Testing

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Summary

In this lesson, you learned:

• Null hypothesis (H₀): No effect/difference; what we test against
• Alternative hypothesis (H₁): Research claim we seek evidence for
• Test statistic: Measures how far sample is from H₀
• P-value: Probability of data this extreme if H₀ true
• Decision rule: Reject H₀ if p ≤ α
• Type I error (α): False positive (rejecting true H₀)
• Type II error (β): False negative (failing to reject false H₀)
• Power = 1 - β: Ability to detect real effects
• Statistical significance ≠ practical importance

Practice Problems

1. State the null and alternative hypotheses for: a) Testing if average commute time differs from 30 minutes b) Testing if a new process reduces defects below 2% c) Testing if customer satisfaction increased above 4.0

2. A sample of 49 has mean 85 and s = 14. Test H₀: μ = 80 vs H₁: μ ≠ 80 at α = 0.05.

3. A z-test yields z = 1.8 for a right-tailed test. a) What is the p-value? b) At α = 0.05, what’s the decision? c) At α = 0.01, what’s the decision?

4. For the battery example, what type of error could we have made with our decision to reject H₀?

Click to see answers

1. a) H₀: μ = 30, H₁: μ ≠ 30 (two-tailed) b) H₀: p ≥ 0.02, H₁: p < 0.02 (left-tailed) c) H₀: μ ≤ 4.0, H₁: μ > 4.0 (right-tailed)

2. t = (85 - 80)/(14/√49) = 5/2 = 2.5 df = 48, critical t₀.₀₂₅ ≈ 2.01 Since |2.5| > 2.01, reject H₀ P-value ≈ 0.016 < 0.05

3. a) P(Z > 1.8) = 1 - 0.9641 = 0.0359 b) 0.0359 < 0.05, reject H₀ c) 0.0359 > 0.01, fail to reject H₀

4. Since we rejected H₀, if we’re wrong, it’s a Type I error (rejected a true H₀). The probability of this error is α = 0.05.

Next Steps

Continue learning about hypothesis testing:

• T-Tests - One-sample and two-sample tests
• Chi-Square Tests - Tests for categorical data
• T-Test Calculator - Practice hypothesis testing