91Ó°ÊÓ

Suppose the population distribution is normal with known \(\sigma\). Let \(\gamma\) be such that \(0<\gamma<\alpha\). For testing \(H_{0}: \mu=\mu_{0}\) versus \(H_{\mathrm{a}}: \mu \neq \mu_{0}\), consider the test that rejects \(H_{0}\) if either \(z \geq z_{\gamma}\) or \(z \leq-z_{\alpha-\gamma}\), where the test statistic is \(Z=\left(\bar{X}-\mu_{0}\right) /\) \((\sigma / \sqrt{n})\) a. Show that \(P\) (type I error) \(=\alpha\). b. Derive an expression for \(\beta\left(\mu^{\prime}\right)\). [Hint: Express the test in the form "reject \(H_{0}\) if either \(\bar{x} \geq c_{1}\) or \(\leq c_{2}\)."] c. Let \(\Delta>0\). For what values of \(\gamma\) (relative to \(\alpha\) ) will \(\beta\left(\mu_{0}+\Delta\right)<\beta\left(\mu_{0}-\Delta\right) ?\)

Step by step solution

01

Understand the Problem

We are asked to investigate a hypothesis test for a normal distribution with known standard deviation, where the rejection regions are given by the critical values \(z \geq z_{\gamma}\) or \(z \leq -z_{\alpha - \gamma}\). We need to determine the probability of a type I error, derive the expression for the power of the test, and decide on conditions where the power is higher for a specific direction.

02

Calculate Type I Error Probability

The type I error probability is the probability of rejecting \(H_0\) when it is true. For a two-tailed test, this probability includes the area in the tails given by the critical values \(z_{\gamma}\) and \(-z_{\alpha-\gamma}\). Thus, the probability of a type I error is given by the sum of the probabilities of the two tail areas: \(P(Z \geq z_{\gamma}) + P(Z \leq -z_{\alpha-\gamma}) = \gamma + (\alpha - \gamma) = \alpha\).

03

Derive Expression for \(\beta(\mu')\)

\(\beta(\mu')\) is the probability of failing to reject \(H_0\) when \(\mu' eq \mu_0\). Recasting the test in terms of \(\bar{x}\), we find cutoffs \(c_1\) and \(c_2\) such that \(\bar{x} \geq c_1\) or \(\bar{x} \leq c_2\). Use \(Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}\) to recalculate in terms of \(\mu'\) to determine \(c_1\) and \(c_2\) based on the standard deviation and critical values. Compute \(P(c_2 < \bar{X} < c_1)\) using \(\Phi\).

04

Analyze Direction of \(\beta\) for \(\gamma\) Values

We need to compare \(\beta(\mu_0 + \Delta)\) and \(\beta(\mu_0 - \Delta)\). The power function has two components depending on where the non-central mean \(\mu'\) lies. If the power is higher for the alternative \(\mu_0 + \Delta\), then more weight should be in the right tail. Thus, \(\gamma > \frac{\alpha}{2}\) ensures \(\beta(\mu_0 + \Delta) < \beta(\mu_0 - \Delta)\), since this favors \(\mu' = \mu_0 + \Delta\).

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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 in hypothesis testing when the null hypothesis, denoted as \( H_0 \), is incorrectly rejected when it is actually true. This is also known as a "false positive." The probability of making a type I error is represented by \( \alpha \), which is the significance level of the test. In the given problem, the significance level, \( \alpha \), represents the total probability of the rejection regions in both tails of the normal distribution.

• Type I error is crucial because it deals with the risks involved in making decisions based on statistical hypothesis testing.
• In a two-tailed test, the type I error probability is split between both ends of the distribution.
• Understand that increasing \( \alpha \) increases the probability of a type I error, reducing the test's reliability.

When configuring a hypothesis test, especially in critical applications, selecting an appropriate \( \alpha \) is crucial to balance the risks of making type I errors against the benefits of detecting a true effect.

Power of a Test

The power of a test, denoted as \( 1 - \beta \), is the probability that the test correctly rejects the null hypothesis \( H_0 \) when the alternative hypothesis \( H_a \) is true. It reflects the test's sensitivity to detect an actual effect.

• Power is vital because it indicates the likelihood of avoiding a type II error (failing to reject \( H_0 \) when \( H_a \) is true).
• A high power, close to 1, means the test is effective in detecting true positives.
• Factors affecting power include sample size, effect size, significance level \( \alpha \), and variance in the data.

In our problem, we derive \( \beta(\mu') \) to understand power dynamics. If the power is required to be higher in one direction (such as \( \mu_0 + \Delta \)), modifications like adjusting \( \gamma \) (where \( \gamma > \frac{\alpha}{2} \)) can be employed to influence test directionality and sensitivity.

Normal Distribution

A normal distribution, also known as a Gaussian distribution, is a continuous probability distribution that is symmetric around the mean. It is characterized by its bell-shaped curve, where most of the data points lie around the mean, with the spread determined by the standard deviation \( \sigma \).

• In hypothesis testing, it is often assumed that the sample data follows a normal distribution, particularly with large sample sizes due to the central limit theorem.
• The properties of a normal distribution, such as symmetry and defined standard deviations from the mean, simplify the calculation of critical values.
• Critical values in a normal distribution determine the rejection regions in hypothesis testing, which are essential for understanding type I errors.

The problem illustrates a setting where normal distribution is the basis, making it easier to calculate probabilities for type I errors and power functions using the standard normal table or z-scores.

Critical Values

Critical values in hypothesis testing are the thresholds at which the null hypothesis \( H_0 \) is rejected. They depend on the significance level \( \alpha \) and the distribution of the test statistic.

• These values define the rejection regions for the test statistic under the null hypothesis.
• In a two-tailed test for a normal distribution, critical values are typically denoted by \( z_{\gamma} \) and \(-z_{\alpha - \gamma} \).
• The proper choice of critical values ensures that the type I error probability remains at the chosen significance level \( \alpha \).

For this exercise, shifts in \( \gamma \) relative to \( \alpha \) influence the positioning of critical values, thereby affecting the balance and directionality of the power in hypothesis testing. By understanding how critical values relate to the area under the normal curve, students can more effectively manage error probabilities in testing scenarios.