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Magazine Chapter 9: Problem 33
Show that for any \(\Delta>0\), when the population distribution is normal and \(\sigma\) is known, the twotailed test satisfies \(\beta\left(\mu_{0}-\Delta\right)=\beta\left(\mu_{0}+\Delta\right)\), so that \(\beta\left(\mu^{\prime}\right)\) is symmetric about \(\mu_{0}\).
Short Answer
Expert verified
The Type II error \( \beta(\mu') \) is symmetric about \( \mu_0 \).
Step by step solution
01
Understand the problem
We need to show that the Type II error probability \( \beta(\mu^\prime) \) for a two-tailed test is symmetric about the hypothesized mean \( \mu_0 \). This means proving \( \beta(\mu_0 - \Delta) = \beta(\mu_0 + \Delta) \). By definition, for a two-tailed test, the rejection region for the null hypothesis \( H_0 : \mu = \mu_0 \) is determined by the critical values on both sides of the sampling distribution based on \( \sigma \) and \( \alpha \).
02
Define the test statistics
For a population with known variance \( \sigma^2 \), the test statistic is \( Z = \frac{\bar{X} - \mu_0}{\sigma/\sqrt{n}} \), where \( \bar{X} \) is the sample mean and \( n \) is the sample size. The null hypothesis is rejected if \( Z \leq -z_{\alpha/2} \) or \( Z \geq z_{\alpha/2} \), where \( z_{\alpha/2} \) is the critical value for a two-tailed test.
03
Calculate the Type II error probability \( \beta(\mu^\prime) \)
For any specific alternative mean \( \mu' \), \( \beta(\mu') \) is the probability that the test fails to reject \( H_0 \). It is given by: \( \beta(\mu') = P(-z_{\alpha/2} < Z' < z_{\alpha/2}) \), where \( Z' = \frac{\bar{X} - \mu'}{\sigma/\sqrt{n}} \).
04
Express \( Z' \) in terms of \( \mu_0 \pm \Delta \)
When \( \mu' = \mu_0 - \Delta \) or \( \mu' = \mu_0 + \Delta \), we calculate the Type II error for two cases. \( Z' \) becomes \( Z' = \frac{\bar{X} - (\mu_0 - \Delta)}{\sigma/\sqrt{n}} \) or \( Z' = \frac{\bar{X} - (\mu_0 + \Delta)}{\sigma/\sqrt{n}} \).
05
Compare probabilities for symmetry
The probability \( P(-z_{\alpha/2} < Z' < z_{\alpha/2}) \) is identical regardless of whether \( \mu' = \mu_0 - \Delta \) or \( \mu' = \mu_0 + \Delta \) because of the symmetry of the normal distribution about \( \mu_0 \). Thus, \( \beta(\mu_0 - \Delta) = \beta(\mu_0 + \Delta) \). It shows that the Type II error probability function \( \beta(\mu') \) is symmetric about \( \mu_0 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Two-tailed test
In statistics, a two-tailed test is a method used in hypothesis testing when we are interested in deviations on either side of a hypothesized parameter value. This means the test examines whether a sample differs significantly from the null hypothesis in either direction. For example, when testing if a population mean is significantly different from a specified value, not just greater or less than the value, a two-tailed test is appropriate.
In a two-tailed test, the rejection region is divided into two tails. This allows us to assess the likelihood of observing an extreme value in either direction. The critical values, which we will discuss further, mark the boundaries of these tails. Thus, when the test statistic falls into either tail, the null hypothesis can be rejected, suggesting a significant effect.
In summary, opting for a two-tailed test is crucial when the research question involves possible deviations in both positive and negative directions from a hypothesized value.
In a two-tailed test, the rejection region is divided into two tails. This allows us to assess the likelihood of observing an extreme value in either direction. The critical values, which we will discuss further, mark the boundaries of these tails. Thus, when the test statistic falls into either tail, the null hypothesis can be rejected, suggesting a significant effect.
In summary, opting for a two-tailed test is crucial when the research question involves possible deviations in both positive and negative directions from a hypothesized value.
Normal distribution
The normal distribution, often called the bell curve, is a continuous probability distribution that is symmetric and centered around its mean. It's one of the most important distributions in statistics due to its natural occurrence in many real-world phenomena.
A key property of the normal distribution is its symmetry about the mean. This means that the probability of deviations above the mean is equal to the probability of equivalent deviations below the mean, which directly ties into the symmetry in the Type II error probability discussed in hypothesis testing.
When a population is normally distributed, statistical methods and predictions become more straightforward due to the well-defined nature of the normal distribution. The spread of data points known as the standard deviation, denoted by \( \sigma \), plays a crucial role in determining the critical values for hypothesis testing, ensuring decisions based on data analysis are precise and reliable.
A key property of the normal distribution is its symmetry about the mean. This means that the probability of deviations above the mean is equal to the probability of equivalent deviations below the mean, which directly ties into the symmetry in the Type II error probability discussed in hypothesis testing.
When a population is normally distributed, statistical methods and predictions become more straightforward due to the well-defined nature of the normal distribution. The spread of data points known as the standard deviation, denoted by \( \sigma \), plays a crucial role in determining the critical values for hypothesis testing, ensuring decisions based on data analysis are precise and reliable.
Critical values
Critical values are the threshold points in hypothesis testing that determine the boundary for the rejection of the null hypothesis. They are based on the chosen significance level \( \alpha \), which represents the probability of making a Type I error, rejecting a true null hypothesis.
For a two-tailed test, the critical values are symmetrically located on both sides of the distribution. These values are crucial because they define what is considered a statistically significant result. If the calculated test statistic exceeds these critical thresholds, the null hypothesis is rejected.
The critical values for a two-tailed test are typically obtained from the standard normal distribution table or a t-distribution table, based on whether the population variance is known or unknown. For a specified \( \alpha \), the critical regions lie at \( -z_{\alpha/2} \) and \( z_{\alpha/2} \) for the normal distribution, encompassing \( \alpha/2 \) in each tail of the distribution.
For a two-tailed test, the critical values are symmetrically located on both sides of the distribution. These values are crucial because they define what is considered a statistically significant result. If the calculated test statistic exceeds these critical thresholds, the null hypothesis is rejected.
The critical values for a two-tailed test are typically obtained from the standard normal distribution table or a t-distribution table, based on whether the population variance is known or unknown. For a specified \( \alpha \), the critical regions lie at \( -z_{\alpha/2} \) and \( z_{\alpha/2} \) for the normal distribution, encompassing \( \alpha/2 \) in each tail of the distribution.
Hypothesis testing
Hypothesis testing is a fundamental process in statistics used to make inferences about population parameters. It involves two hypotheses: the null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \). The null hypothesis typically states there is no effect or difference, while the alternative suggests the presence of an effect or difference.
The objective in hypothesis testing is to determine whether there is enough statistical evidence in a sample to infer that the null hypothesis should be rejected in favor of the alternative hypothesis. The procedure includes selecting an appropriate test statistic, establishing critical values, and using these values to decide whether to reject or fail to reject the null hypothesis.
Type II error, denoted by \( \beta \), is an important concept here. It represents the probability of failing to reject a false null hypothesis. Understanding this error and ensuring \( \beta \) is minimized is critical for reliable hypothesis testing. Furthermore, the symmetry of \( \beta(\mu') \) around the mean \( \mu_0 \) stems from the symmetry of the normal distribution, offering deeper insights into the error probabilities across the distribution.
The objective in hypothesis testing is to determine whether there is enough statistical evidence in a sample to infer that the null hypothesis should be rejected in favor of the alternative hypothesis. The procedure includes selecting an appropriate test statistic, establishing critical values, and using these values to decide whether to reject or fail to reject the null hypothesis.
Type II error, denoted by \( \beta \), is an important concept here. It represents the probability of failing to reject a false null hypothesis. Understanding this error and ensuring \( \beta \) is minimized is critical for reliable hypothesis testing. Furthermore, the symmetry of \( \beta(\mu') \) around the mean \( \mu_0 \) stems from the symmetry of the normal distribution, offering deeper insights into the error probabilities across the distribution.
