Chapter 4: Problem 11
Let \(Y_{1}
Short Answer
Expert verified
(a) The critical value \( c \) is 0.84. (b) The power function for a given \( \theta \) is \( \beta(\theta) = 1 - (\frac{c}{\theta})^{4} \)
Step by step solution
01
Understanding the Hypothesis Test
The hypothesis test is defined as:\n Null Hypothesis \( H_{0}: \theta=1 \) \n Alternative Hypothesis \( H_{1}: \theta>1 \)\n The test statistic is \(Y_{4}\), the largest order statistic, and it is compared to a threshold \(c\) such that if \(Y_{4} \geq c\), we reject \( H_{0} \) in favor of \( H_{1} \).
02
Deriving the Distribution of \( Y_{4} \) under \( H_{0} \)
Under the null hypothesis (\( H_{0}: \theta=1 \)), \( Y_{4} \) has a Beta distribution with parameters \( n = 4 \) and \( k = 4 \) because for any \( x \), \( P(Y_{4} \leq x) = [F(x)]^{n} = [x^k]^{n} = x^{nk} \) when \( 0 \leq x \leq 1 \). So \( Y_{4} \) has a beta distribution \( Beta(4, 4) \).
03
Finding the Critical Value \( c \)
We need to set the significance level to \(\alpha = 0.05 \). Looking at how the power function will be expected to increase with \( n \), it can be said that the rejection region is of the form \( Y_{4} \geq c \). This would translate to \( P(Y_{4} \geq c) = \alpha \). Using the CDF of \( Beta(4,4) \) and setting it equal to \( 1 - \alpha \), solve for \( c \). By using distribution tables or computational tools, this yields \( c = 0.84 \).
04
Determine the Power Function of the Test
The power function is equal to the probability of rejecting \( H_{0} \) when it is false, i.e., \( H_{1} \) is true. Thus, we want \( P(Y_{4} \geq c | \theta ) \). The power function for a given \( \theta \) will be \( \beta(\theta) = 1 - (\frac{c}{\theta})^{4} \). The power of the test increases as \( \theta \) increases which is desired in a good test.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
pdf (probability density function)
When working with continuous random variables, the probability density function (pdf) plays a vital role in defining the likelihood of a value occurring within a particular range. Unlike a probability mass function which provides probabilities for discrete random variables, a pdf gives us the relative 'density' of probabilities across a range of values. In the hypothesis testing problem provided, we encounter the pdf defined by \( f(x ; \theta)=\frac{1}{\theta} \), for \( 0
This pdf specifies a uniform distribution over the interval from 0 to \( \theta \), meaning that the variable represented has an equal probability of falling within any interval of equal length within its prescribed range. Understanding the pdf is essential to compute the likelihoods required for hypothesis testing, where the selection of critical values and the calculation of probabilities depend on its characteristics.
The area under the pdf curve over an interval corresponds to the probability of the variable falling within that interval. Therefore, calculating the total area under the pdf between two points gives the probability that the random variable is between those points, which is crucial for determining the rejection region of a hypothesis test.
This pdf specifies a uniform distribution over the interval from 0 to \( \theta \), meaning that the variable represented has an equal probability of falling within any interval of equal length within its prescribed range. Understanding the pdf is essential to compute the likelihoods required for hypothesis testing, where the selection of critical values and the calculation of probabilities depend on its characteristics.
The area under the pdf curve over an interval corresponds to the probability of the variable falling within that interval. Therefore, calculating the total area under the pdf between two points gives the probability that the random variable is between those points, which is crucial for determining the rejection region of a hypothesis test.
order statistics
Order statistics provide a way to organize sample data and identify distributional properties within a dataset, such as minimums, maximums, medians, and other quantiles. The order statistics of a sample are the values in the sample arranged in ascending order. In our problem, \( Y_{1} < Y_{2} < Y_{3} < Y_{4} \) are the order statistics for a random sample of size 4.
Importantly, the largest order statistic, \( Y_{4} \) in the given set, serves as a pivotal element in hypothesis testing. This value represents the upper bound of our sample data and is used as a test statistic to decide whether to reject the null hypothesis, \( H_{0} \), based on its position relative to a critical value, \( c \), which is predefined according to a chosen significance level.
Understanding order statistics is crucial because they have their own distributions, which must be accounted for when performing hypothesis tests. The distribution of \( Y_{4} \), as derived in the exercise, is essential for finding the threshold value that delineates the acceptance or rejection region for our test.
Importantly, the largest order statistic, \( Y_{4} \) in the given set, serves as a pivotal element in hypothesis testing. This value represents the upper bound of our sample data and is used as a test statistic to decide whether to reject the null hypothesis, \( H_{0} \), based on its position relative to a critical value, \( c \), which is predefined according to a chosen significance level.
Understanding order statistics is crucial because they have their own distributions, which must be accounted for when performing hypothesis tests. The distribution of \( Y_{4} \), as derived in the exercise, is essential for finding the threshold value that delineates the acceptance or rejection region for our test.
significance level
The significance level, denoted as \( \alpha \), is a threshold that helps to determine the criteria for rejecting the null hypothesis, \( H_{0} \), in a hypothesis test. Usually set before the test is conducted, the significance level quantifies the risk of making a type I error — that is, the error of rejecting a true null hypothesis.
In our exercise, a significance level of 0.05, or 5%, was chosen. This indicates that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis. To apply this to the problem, we must find the critical value \( c \) such that there is only a 5% probability that \( Y_{4} \), the maximum order statistic, will exceed \( c \) when the null hypothesis is true. The value of \( c \) essentially establishes the cutoff point beyond which we will consider the observed results to be statistically significant and indicative of the alternative hypothesis, \( H_{1} \), being true.
Choosing an appropriate significance level is subject to the context of the study, as it balances the chance of making a type I error against the need to avoid a type II error, where the test fails to reject a false null hypothesis. Ensuring that students grasp this concept of a significance level greatly enhances their understanding of decision-making under uncertainty in statistical hypothesis testing.
In our exercise, a significance level of 0.05, or 5%, was chosen. This indicates that we are willing to accept a 5% chance of incorrectly rejecting the null hypothesis. To apply this to the problem, we must find the critical value \( c \) such that there is only a 5% probability that \( Y_{4} \), the maximum order statistic, will exceed \( c \) when the null hypothesis is true. The value of \( c \) essentially establishes the cutoff point beyond which we will consider the observed results to be statistically significant and indicative of the alternative hypothesis, \( H_{1} \), being true.
Choosing an appropriate significance level is subject to the context of the study, as it balances the chance of making a type I error against the need to avoid a type II error, where the test fails to reject a false null hypothesis. Ensuring that students grasp this concept of a significance level greatly enhances their understanding of decision-making under uncertainty in statistical hypothesis testing.
power function of the test
In statistical hypothesis testing, the power function of the test, represented by \( \beta(\theta) \), measures the test's ability to correctly reject a false null hypothesis. The power is the probability that the test statistic falls into the rejection region given that the alternative hypothesis is true. When the null hypothesis is false, a test with high power is more likely to detect this and correctly reject it.
In the given exercise, determining the power function involves understanding how the test statistic, \( Y_{4} \), behaves under different values of \( \theta \), when \( \theta > 1 \). The power function we derived is \( \beta(\theta) = 1 - (\frac{c}{\theta})^{4} \), where \( c \) is the critical value. This function tells us that as \( \theta \) increases, the probability of observing a value of \( Y_{4} \) at least as large as \( c \) also increases, indicating that the test becomes more powerful against larger true values of \( \theta \).
Understanding the power function is important since it helps to evaluate the performance of a hypothesis test. A well-designed test aims to have high power, which translates into a high probability of detecting an effect or difference when one exists, thus minimizing the chance of a type II error.
In the given exercise, determining the power function involves understanding how the test statistic, \( Y_{4} \), behaves under different values of \( \theta \), when \( \theta > 1 \). The power function we derived is \( \beta(\theta) = 1 - (\frac{c}{\theta})^{4} \), where \( c \) is the critical value. This function tells us that as \( \theta \) increases, the probability of observing a value of \( Y_{4} \) at least as large as \( c \) also increases, indicating that the test becomes more powerful against larger true values of \( \theta \).
Understanding the power function is important since it helps to evaluate the performance of a hypothesis test. A well-designed test aims to have high power, which translates into a high probability of detecting an effect or difference when one exists, thus minimizing the chance of a type II error.
Beta distribution
The Beta distribution is a versatile distribution that is particularly useful in situations where a probability is modeled for events that have two possible outcomes. It is commonly used in Bayesian statistics, reliability engineering, and hypothesis testing. The Beta distribution is characterized by two shape parameters, \( \alpha \) and \( \beta \), which influence its form - making it flexible to model a range of different scenarios.
In the context of our exercise, the Beta distribution arises naturally as the distribution of the order statistic \( Y_{4} \) under the null hypothesis that \( \theta=1 \). Since the individual observations are uniformly distributed, it can be shown theoretically that the \( k \)-th order statistic follows a Beta distribution with parameters \( k \) and \( n-k+1 \). Hence, \( Y_{4} \) has a \( Beta(4, 1) \) distribution when \( \theta=1 \).
Knowledge of the Beta distribution is paramount, as we exploit its cumulative distribution function (CDF) to solve for the critical value \( c \), beyond which the test rejects the null hypothesis. Familiarization with the Beta distribution's properties and its role in hypothesis testing underlines its conceptual importance in statistics and helps students apply it to a variety of statistical challenges.
In the context of our exercise, the Beta distribution arises naturally as the distribution of the order statistic \( Y_{4} \) under the null hypothesis that \( \theta=1 \). Since the individual observations are uniformly distributed, it can be shown theoretically that the \( k \)-th order statistic follows a Beta distribution with parameters \( k \) and \( n-k+1 \). Hence, \( Y_{4} \) has a \( Beta(4, 1) \) distribution when \( \theta=1 \).
Knowledge of the Beta distribution is paramount, as we exploit its cumulative distribution function (CDF) to solve for the critical value \( c \), beyond which the test rejects the null hypothesis. Familiarization with the Beta distribution's properties and its role in hypothesis testing underlines its conceptual importance in statistics and helps students apply it to a variety of statistical challenges.
