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.

Concept Check 🎯

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 $\alpha$

Concept Check 🎯

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.

Concept Check 🎯

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

A larger sample makes the standard error smaller ($SE = s/\sqrt{n}$), so the sampling distribution of the statistic is narrower. A real difference between the true parameter and the null value then stands out more clearly, so power increases. The gain is not proportional: because SE depends on $\sqrt{n}$, you need four times the sample to halve the SE.

Size of the True Difference

The farther the true parameter is from the null value (for example, a true mean of 110 instead of 106 when $H_0: \mu = 100$), the easier the difference is to detect, so power is higher. Less variability in the population ($\sigma$ smaller) has the same effect.

ℹ️ The AP exam asks how these factors change power; it does not ask you to calculate 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.

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

Concept Check 🎯

Power Analysis 🧮

$H_0: \mu = 100$, $\sigma = 15$, same $n$ and $\alpha$. Compare two possible true means: $\mu = 106$ and $\mu = 110$.

1) How far is $\mu = 106$ from the null value?

2) Which true mean gives the test more power? (106/110)

3) To increase power at $\mu = 106$, 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. Choose a sample size large enough to give that power

Example

A school wants to know whether a new reading program raises mean test scores by at least 5 points. Scores vary a lot from student to student ($\sigma \approx 20$), so a 5-point gain is small compared with the spread. A study of 15 students per group would have low power: even if the program works, the test would probably fail to reject $H_0$. Planning a much larger study raises the power to an acceptable level (often 0.80). Statistical software finds the exact sample size; the AP exam asks only for the reasoning.

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.”

Concept Check 🎯

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)

Concept Check 🎯

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

Concept Check 🎯

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.