Question

Type I and Type II Errors. In Exercises 29–32, 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 people who require no vision correction is less than 0.25.

Short answer

A type I error occurs when the actual value of the proportion is equal to 0.25, and the researcher rejects the claim\(p = 0.25\)and supports the claim\(p < 0.25\).

A type II error occurs when the actual value of the proportion is less than 0.25, and the researcher fails to reject the claim \(p = 0.25\).

Step-by-step solution

Given information

It is claimed that the proportion of people who require no vision correction is less than 0.25.

Hypotheses

Let p be the population proportion of people who require no vision correction.

According to the stated claim, the following hypotheses are set up:

Null hypothesis \({H_0}:p = 0.25\).

Alternative hypothesis \({H_A}:p < 0.25\).

Types of errors

The two types of errors made while conducting hypotheses tests are defined below.

Type I error: Rejecting the null hypothesis when the null hypothesis is true is a type I error and is denoted by\(\alpha \).

Type II error: Failing to reject the null hypothesis when the null hypothesis is false is a type II error and is denoted by\(\beta \).

In accordance with the given claim, the following statements define the type I error and the type II error:

Type I error: When the actual value of the proportion is equal to 0.25, and the researcher rejects the claim \(p = 0.25\) and supports the claim \(p < 0.25\), a type I error is made.

Type II error: When the actual value of the proportion is less than 0.25, and the researcher fails to reject the claim \(p = 0.25\), a type II error is made.