91Ó°ÊÓ

Let \(X_{1}, \ldots, X_{n}\) denote a random sample from a normal population distribution with a known value of \(\sigma\). a. For testing the hypotheses \(H_{0}: \mu=\mu_{0}\) versus \(H_{\mathrm{a}}: \mu>\mu_{0}\) (where \(\mu_{0}\) is a fixed number), show that the test with test statistic \(\bar{X}\) and rejection region \(\bar{x} \geq \mu_{0}+2.33 \sigma / \sqrt{n}\) has significance level .01. b. Suppose the procedure of part (a) is used to test \(H_{0}: \mu \leq \mu_{0}\) versus \(H_{\mathrm{a}}: \mu>\mu_{0}\). If \(\mu_{0}=100\), \(n=25\), and \(\sigma=5\), what is the probability of committing a type I error when \(\mu=99\) ? When \(\mu=98\) ? In general, what can be said about the probability of a type I error when the actual value of \(\mu\) is less than \(\mu_{0}\) ? Verify your assertion.

Step by step solution

01

Understanding the Hypotheses

We are testing the hypothesis \( H_0: \mu = \mu_0 \) against the alternative \( H_a: \mu > \mu_0 \). Here, \( \mu_0 \) is a specified value, and we want to determine at what point to reject the null hypothesis based on the sample mean \( \bar{X} \).

02

Calculating the Rejection Region

For the test with significance level \( \alpha = 0.01 \), we need to find the critical value from the standard normal distribution corresponding to \( 1 - \alpha = 0.99 \), which is approximately 2.33. Thus, we reject \( H_0 \) if \( \bar{x} \geq \mu_0 + 2.33 \frac{\sigma}{\sqrt{n}} \).

03

Evaluating Part (b) Test Setup

In this part, we'll calculate the probability of a Type I error with the true mean not being equal to \( \mu_0 \), specifically for \( \mu = 99 \) and \( \mu = 98 \). However, a Type I error occurs only when \( H_0 \) is true, so if \( \mu \leq \mu_0 \) and we reject \( H_0 \), that's a Type I error condition.

04

Calculating Type I Error Probability for Specific \( \mu \) Values

Although the Type I error occurs when \( \mu = \mu_0 \) and \( H_0 \) is falsely rejected, when actual \( \mu \) values are less than \( \mu_0 \), actions are conservative as \( \bar{x} \) will be less likely to exceed \( \mu_0 + 2.33 \frac{\sigma}{\sqrt{n}} \). Thus, the probability of erroneously rejecting \( H_0 \) when \( \mu = 99 \) or \( \mu = 98 \) is actually 0.

05

General Assertion about \( \mu < \mu_0 \)

When the real mean \( \mu \) is less than \( \mu_0 \), the probability of committing a Type I error is essentially zero. This is because the statistical test doesn't favor rejection of \( H_0 \), as the sample mean typically won't satisfy the rejection region condition.

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Key Concepts

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

Type I Error

In hypothesis testing, a **Type I Error** occurs when we wrongly reject a null hypothesis that is actually true. This type of error can be thought of like a false alarm. Imagine you have a fire alarm that rings even when there's no fire. That's similar to a Type I error.

To explain further, suppose we're testing if a factory's machine produces bags with an average weight of exactly 10 kilograms, (\(H_0: \mu = 10\)), against the possibility that they could be heavier (\(H_a: \mu > 10\)). If we conclude bags are heavier when they're not, we've made a Type I error.

• **Occurs when**: The null hypothesis (\(H_0\)) is true, but we reject it.
• **Example**: Telling someone they failed a test when they actually passed.
• **Consequence**: May lead to unnecessary actions like recalibrating machines needlessly.

Understanding Type I Errors is crucial for designing tests with appropriate significance levels to minimize these errors.

Significance Level

The **Significance Level**, often denoted as \( \alpha \), is a threshold set by the researcher which determines under what criteria the null hypothesis will be rejected. It's the risk of making a Type I error you are willing to take when performing a hypothesis test.

For example, a significance level of 0.01 means there's a 1% risk you're committing a Type I error. In practical terms, when you conclude something is significant, there's only a 1% chance you're wrong if the null hypothesis is actually true.

• **Common values**: 0.01, 0.05, 0.10.
• **Dictates the test's critical value** based on the distribution used.
• **Testing Process: ** Choose a significance level, compare your test statistic to the corresponding critical value.

By setting a lower significance level, you become more conservative, reducing the chance of Type I errors at the cost of potentially not detecting true effects, known as Type II errors.

Normal Distribution

**Normal Distribution** plays a fundamental role in hypothesis testing, especially when dealing with continuous data. It is a symmetric, bell-shaped curve characterized by its mean (\(\mu\)) and standard deviation (\(\sigma\)).

In hypothesis testing, especially for tests concerning means, we often assume the sample data comes from a normal distribution.

• **Bell-shaped curve**: Most of the data falls around the mean.
• **Symmetric**: Left and right halves are mirror images.
• **Central Limit Theorem**: With a large enough sample size, sample means will approximate a normal distribution, even if the data itself is not normal.

Understanding how normal distribution works help you apply the correct tests, determine critical values, and make more accurate inferences from your data. It also allows for more robust tests even when some data assumptions are slightly violated.