Type I and Type II Errors
In Exercises 25–28, provide statements that identify the type I error and the type II error that correspond to the given claim. (Although conclusions are usually expressed in verbal form, the answers here can be expressed with statements that include symbolic expressions such as p = 0.1.)


The proportion of drivers who make angry gestures is greater than 0.25.

Verified step by step guidance

1. Step 1: Understand the claim. The claim states that the proportion of drivers who make angry gestures is greater than 0.25. Symbolically, this can be written as H₁: p > 0.25, where p represents the true proportion of drivers who make angry gestures.
2. Step 2: Define the null hypothesis. The null hypothesis (H₀) is the opposite of the claim. It states that the proportion of drivers who make angry gestures is less than or equal to 0.25. Symbolically, H₀: p ≤ 0.25.
3. Step 3: Define a Type I error. A Type I error occurs when the null hypothesis is rejected even though it is true. In this context, a Type I error would mean concluding that the proportion of drivers who make angry gestures is greater than 0.25 (accepting H₁) when in reality, it is less than or equal to 0.25.
4. Step 4: Define a Type II error. A Type II error occurs when the null hypothesis is not rejected even though it is false. In this context, a Type II error would mean failing to conclude that the proportion of drivers who make angry gestures is greater than 0.25 (failing to accept H₁) when in reality, it is greater than 0.25.
5. Step 5: Summarize the errors. Type I error: Concluding p > 0.25 when p ≤ 0.25. Type II error: Failing to conclude p > 0.25 when p > 0.25.

Key Concepts

Type I Error

A Type I error occurs when a true null hypothesis is incorrectly rejected. In the context of the given claim, it would mean concluding that the proportion of drivers who make angry gestures is greater than 0.25 when, in fact, it is not. This error represents a false positive, leading to the incorrect belief that there is an effect or difference when there is none.

Type II Error

A Type II error happens when a false null hypothesis is not rejected. For the claim regarding the proportion of drivers making angry gestures, this would mean failing to conclude that the proportion is greater than 0.25 when it actually is. This error represents a false negative, resulting in the missed opportunity to identify a true effect or difference.

Null Hypothesis and Alternative Hypothesis

In hypothesis testing, the null hypothesis (H0) represents a statement of no effect or no difference, while the alternative hypothesis (H1) suggests that there is an effect or difference. For the claim about drivers making angry gestures, the null hypothesis would state that the proportion is less than or equal to 0.25, while the alternative hypothesis would assert that it is greater than 0.25. Understanding these hypotheses is crucial for identifying Type I and Type II errors.