Type I and Type II Errors

⚠️ Type I and Type II Errors

Part 1 of 7 — Error Types

Two Kinds of Errors

Definitions

• Type I Error: Rejecting $H_0$ when it’s actually true (false positive)
  • Probability = $\\alpha$ (significance level)
• Type II Error: Failing to reject $H_0$ when it’s actually false (false negative)
  • Probability = $\\beta$

Analogy

🔑 The significance level $\\alpha$ is the probability of a Type I error. YOU choose $\\alpha$ before the test.

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Error Classification 🧮

$H_0$: The defendant is innocent. $H_a$: The defendant is guilty.

1) A Type I error in this context means: convicting an _______ person. (innocent/guilty)

2) A Type II error means: acquitting a _______ person. (innocent/guilty)

3) If $\\alpha = 0.01$, the probability of wrongly convicting an innocent person is ___.

📊 Significance Level and Errors

Part 2 of 7 — The Tradeoff

The $\\alpha$/$\\beta$ Tradeoff

Decreasing $\\alpha$ (harder to reject $H_0$) → increases $\\beta$ (more likely to miss real effects).

$\\alpha \\downarrow \\implies \\beta \\uparrow$

You can’t minimize both simultaneously with a fixed sample size.

Choosing $\\alpha$

Consequences Matter

Choose $\\alpha$ by considering which error is worse:

• If Type I is more serious → use smaller $\\alpha$
• If Type II is more serious → use larger $\\alpha$

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Error Tradeoff 🧮

1) If $\\alpha = 0.05$, what is the probability of a Type I error?

2) Lowering $\\alpha$ to 0.01 makes $\\beta$ go ___ (up/down).

3) What must increase to reduce BOTH types of errors simultaneously?

💪 Power of a Test

Part 3 of 7 — Detecting Real Effects

What Is Power?

$\\text{Power} = 1 - \\beta = P(\\text{reject } H_0 \\mid H_0 \\text{ is false})$

Power is the probability of correctly detecting a real effect.

Desirable Power

• Power of 0.80 (80%) or higher is considered adequate
• Power of 0.90 (90%) is preferred in many studies

What Affects Power?

Visual Interpretation

Power relates to the overlap between the null distribution and the true distribution. Less overlap → more power.

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Power Calculations 🧮

A test has $\\beta = 0.25$.

1) What is the power of this test?

2) Is this power considered adequate by the 80% standard? (yes/no)

3) Name one way to increase power without changing $\\alpha$.

🔧 Factors Affecting Power

Part 4 of 7 — Detailed Analysis

Sample Size and Power

$\\text{Power} \\propto \\sqrt{n}$

Doubling the sample size increases power, but the relationship is not linear.

Effect Size

The effect size measures the magnitude of the true difference:

$d = \\frac{|\\mu_{\\text{true}} - \\mu_0|}{\\sigma}$

Larger effects are easier to detect → higher power.

One-Tailed vs. Two-Tailed

A one-tailed test has more power than a two-tailed test at the same $\\alpha$, because all $\\alpha$ is concentrated in one tail.

$\\text{Power}_{\\text{one-tail}} > \\text{Power}_{\\text{two-tail}}$

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Power Analysis 🧮

$H_0: \\mu = 100$. True $\\mu = 106$, $\\sigma = 15$.

1) Effect size $d = |106 - 100|/15 = ?$ (simplified)

2) Is this a small, medium, or large effect?

3) To increase power, should you increase or decrease $n$?

💡 Applications

Part 5 of 7 — Real-World Power Analysis

Power Analysis in Study Design

Before collecting data, researchers should:

1. Specify the smallest effect worth detecting
2. Choose $\\alpha$ and desired power (often 0.80)
3. Calculate the required sample size

Example

Want to detect a 5-point difference in test scores ($\\sigma = 20$). With $\\alpha = 0.05$ and power = 0.80:

$n \\approx \\left(\\frac{z_{\\alpha/2} + z_{\\beta}}{d}\\right)^2 = \\left(\\frac{1.96 + 0.84}{0.25}\\right)^2 \\approx 126 \\text{ per group}$

Consequences of Low Power

• Wasted resources on a study that likely won’t find anything
• Failure to detect important effects
• Non-significant results that are inconclusive (not evidence of no effect!)

“Absence of evidence is not evidence of absence.”

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Power Application 🧮

A researcher has power = 0.60 and doesn’t reject $H_0$.

1) How likely was the test to detect a real effect?

2) What is $\\beta$?

3) Should the researcher conclude the effect doesn’t exist? (yes/no)

🏆 Problem-Solving Workshop

Part 6 of 7 — AP-Style Practice

Common AP Questions

1. Describe Type I and Type II errors in context
2. Explain the consequences of each error type
3. State which error is more serious and why
4. Relationship between $\\alpha$, $\\beta$, power, and sample size

Template: Describing Errors in Context

Type I: “We conclude [Ha in context] when in reality [H0 in context].”

Type II: “We fail to conclude [Ha in context] when in reality [Ha IS true].”

Example

$H_0$: The new drug is not effective. $H_a$: The new drug IS effective.

• Type I: We conclude the drug is effective when it actually ISN’T (leads to approving an ineffective drug)
• Type II: We fail to detect that the drug IS effective (miss a good drug)

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Error Context 🧮

$H_0$: The parachute meets safety standards. $H_a$: The parachute does NOT meet safety standards.

1) Type I error: reject a parachute that actually _____ safety standards. (meets/fails)

2) Type II error: accept a parachute that actually _____ safety standards. (meets/fails)

3) Which error is more dangerous? (Type I/Type II)

📝 Mixed Review

Part 7 of 7 — Comprehensive Review

Complete Summary

AP Exam Checklist

• Define both error types in context
• State consequences of each error
• Identify which error is more serious
• Know factors that affect power
• Understand the $\\alpha$/$\\beta$ tradeoff

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Final Challenge 🧮

1) $\\alpha = 0.05$, Power = 0.80. What is $\\beta$?

2) True or False: Increasing $n$ always increases $\\alpha$.

3) A test with $\\alpha = 0.05$ incorrectly rejects $H_0$ about ___% of the time when $H_0$ is true.