91Ó°ÊÓ

For each of the following situations, state whether a Type I, a Type II, or neither error has been made. a. A test of \(\mathrm{H}_{0}: \mu=25\) vs. \(\mathrm{H}_{\mathrm{A}}: \mu>25\) rejects the null hypothesis. Later it is discovered that \(\mu=24.9\). b. A test of \(\mathrm{H}_{0}: p=0.8\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.8\) fails to reject the null hypothesis. Later it is discovered that \(p=0.9\). c. A test of \(\mathrm{H}_{0}: p=0.5\) vs. \(\mathrm{H}_{\mathrm{A}}: p \neq 0.5\) rejects the null hypothesis. Later it is discovered that \(p=0.65\). d. A test of \(\mathrm{H}_{0}: p=0.7\) vs. \(\mathrm{H}_{\mathrm{A}}: p<0.7\) fails to reject the null hypothesis. Later it is discovered that \(p=0.6\).

Step by step solution

01

Situation a: Identifying the Error Type

In the first situation, the null hypothesis \(\mathrm{H}_{0}: \mu=25\) was rejected, but it was later discovered that \(\mu=24.9\). Here, a Type I error was committed. This occurs when the null hypothesis is true, but is incorrectly rejected.

02

Situation b: Identifying the Error Type

In situation b, the null hypothesis \(\mathrm{H}_{0}: p=0.8\) was not rejected, and it was later discovered that \(p=0.9\). In this case, no error was made because the null hypothesis was correctly not rejected, as \(p>0.8\).

03

Situation c: Identifying the Error Type

In situation c, the null hypothesis \(\mathrm{H}_{0}: p=0.5\) was rejected, and it was later discovered that \(p=0.65\). Here, no error was made. The null hypothesis was correctly rejected, as \(p \neq 0.5\).

04

Situation d: Identifying the Error Type

In the last situation, the null hypothesis \(\mathrm{H}_{0}: p=0.7\) was not rejected, and it was later discovered that \(p=0.6\). Here, a Type II error was made. This occurs when the null hypothesis is false, but is not rejected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Type I Error

A Type I error occurs when the null hypothesis is true, but it is mistakenly rejected. Imagine you're a teacher grading a test and you mark a correct answer as wrong. That's akin to a Type I error.

In hypothesis testing, we deal with a probability, known as the significance level (denoted by \( \alpha \)). This is the chance of making a Type I error—rejecting a true hypothesis.

• If \( \alpha = 0.05 \), there's a 5% risk of rejecting the null hypothesis when it is actually true.
• This is also referred to as a "false positive"—thinking there's an effect when there's not.

An example from the original exercise is part (a), where the test mistakenly rejected the null hypothesis \( H_0: \mu = 25 \), even though \( \mu = 24.9 \), which is within statistical tolerance of 25.

Type II Error

A Type II error occurs when the null hypothesis is false, but we fail to reject it. This is similar to a situation where someone cheats on an exam, but the teacher doesn't catch it.

In hypothesis testing, this error is represented by \( \beta \), the probability of not rejecting a false null hypothesis.

• If \( \beta \) is too high, it means your test isn't sensitive enough, risking a "false negative"—missing a true effect.
• Reducing \( \beta \) increases the power of the test, allowing you to better detect true effects.

A classic example is situation (d) from the exercise. Here, the hypothesis \( H_0: p = 0.7 \) was not rejected, even though the true proportion was \( p = 0.6 \).

Null Hypothesis

The null hypothesis is a fundamental concept in statistical testing, representing the default position that there is no effect or difference. It is denoted as \( H_0 \).

It's like assuming a person is innocent until proven guilty. Hypothesis tests are designed to test the strength of the evidence against \( H_0 \).

• The null hypothesis is typically set up to be rejected, so evidence against it needs to be strong enough—beyond predetermined significance levels like 0.05 or 0.01.
• Failing to reject \( H_0 \) means that the evidence is not strong enough to support an alternative conclusion.

Throughout the given exercise, we saw several hypotheses like \( \mathrm{H}_0: \mu = 25 \) or \( \mathrm{H}_0: p = 0.7 \). The outcomes depended on whether these null hypotheses were valid or false upon later investigation.