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Chapter 10: Problem 102

Let \(Y_{1}, Y_{2}, \ldots, Y_{n}\) denote a random sample from a Bernoulli- distributed population with parameter p. That is, $$ p\left(y_{i} | p\right)=p^{y_{1}}(1-p)^{1-y_{i}}, \quad y_{i}=0,1 $$ a. Suppose that we are interested in testing \(H_{0}: p=p_{0}\) versus \(H_{a}: p=p_{a},\) where \(p_{0}k^{*}\) for some constant \(k\) iii. Give the rejection region for the most powerful test of \(H_{0}\) versus \(H_{a}\) b. Recall that \(\sum_{i=1}^{n} Y_{i}\) has a binomial distribution with parameters \(n\) and \(p\). Indicate how to determine the values of any constants contained in the rejection region derived in part [a(iii)]. c. Is the test derived in part (a) uniformly most powerful for testing \(H_{0}: p=p_{0}\) versus \(H_{a}: p>p_{0} ?\) Why or why not?

Short Answer

Expert verified
The test is most powerful for specific \( p_a \) but not uniformly most powerful for all \( p > p_0 \).

Step by step solution

01

Likelihood Functions of Bernoulli Distribution

The likelihood function for a Bernoulli-distributed sample \( Y_1, Y_2, \ldots, Y_n \) where each \( Y_i \) is a Bernoulli trial with probability parameter \( p \), is given by:\[ L(p) = \prod_{i=1}^{n} p^{y_i}(1-p)^{1-y_i} = p^{\sum y_i} (1-p)^{n-\sum y_i} \]This is the likelihood function for the entire sample.
02

Ratio of Likelihoods

To compare the two hypotheses, we calculate the ratio of the likelihoods.The hypothesis \( H_0 \, : \, p = p_0 \) has a likelihood function \( L(p_0) \), and the hypothesis \( H_a \, : \, p = p_a \) has a likelihood function \( L(p_a) \). Thus, \[ \frac{L(p_0)}{L(p_a)} = \frac{p_0^{\sum y_i} (1-p_0)^{n-\sum y_i}}{p_a^{\sum y_i} (1-p_a)^{n-\sum y_i}} \]This simplifies to:\[ \left(\frac{p_0}{p_a}\right)^{\sum y_i} \left(\frac{1-p_0}{1-p_a}\right)^{n-\sum y_i} \]Using the identity for exponent simplification:\[ = \left(\frac{p_0(1-p_a)}{(1-p_0)p_a}\right)^{\sum y_i} \left(\frac{1-p_0}{1-p_a}\right)^n \]
03

Inference on Likelihood Ratio

The likelihood ratio test is: reject \(H_0\) if the log likelihood ratio \( \sum_{i=1}^{n} y_{i} > k^{*} \), where \( k^{*} \) is derived from constant \( k \). The inequality \( \frac{L(p_0)}{L(p_a)} < k \) indicates preference for \(H_a\) over \(H_0\), meaning there exists some threshold of the sum \( \sum y_i \) that defines the rejection region.
04

Rejection Region for Most Powerful Test

The most powerful test against an alternative hypothesis involves the likelihood ratio, rejecting \(H_0\) when \(\sum_{i=1}^{n} y_i > k^{*}\) where \( k^{*} \) is chosen based on the specification or level of significance of the test. This effectively means comparing the observed \(\sum y_i\) against a threshold determined by \(k^{*}\).
05

Binomial Distribution and Constants

The test statistic \( \sum_{i=1}^{n} Y_{i} \) follows a binomial distribution \( B(n, p) \). To find \( k^{*} \), identify it from the binomial distribution table or calculate it based on a desired level of significance \(\alpha\). The critical value \( k^{*} \) balances the rejection probability when considering \(H_a: p=p_a\).
06

Uniformly Most Powerful Test

The derived test is the most powerful test for the simple hypothesis \( H_0: p = p_0 \) versus the specific alternative \( H_a: p = p_a \), but not uniformly so for all \( p > p_0 \). In a general scenario where \( H_a: p > p_0 \), the test remains most powerful only for specific \( p_a \), not for a range of \( p > p_0 \). This is why it isn't uniformly most powerful.

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Key Concepts

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

Likelihood ratio test
The likelihood ratio test is a powerful statistical method used to compare two competing hypotheses. It involves comparing the likelihoods of data under different hypotheses. In hypothesis testing, you have a null hypothesis, denoted as \(H_0\), and an alternative hypothesis, \(H_a\). The test is based on calculating the ratio of likelihoods: the likelihood of the data under \(H_0\) divided by the likelihood under \(H_a\).
For a given dataset, this ratio helps determine which hypothesis is more likely. If the likelihood ratio is low, it suggests that the data fits \(H_a\) better than \(H_0\). A crucial part of the test is establishing a threshold value or constant, \(k\), to compare the likelihood ratio against. If the ratio is lower than \(k\), \(H_0\) is rejected in favor of \(H_a\).
  • The advantage of the likelihood ratio test is its applicability, especially in complex models with multiple parameters.
  • It provides a framework for determining the most powerful test for a given significance level.
Bernoulli distribution
The Bernoulli distribution is one of the simplest probability distributions, representing a single experiment with two possible outcomes: success (usually coded as 1) and failure (coded as 0). It is parameterized by \(p\), the probability of success.
Each trial in a Bernoulli process involves the same probability of success, leading to the Bernoulli probability function:\[ p(y_i|p) = p^{y_i}(1-p)^{1-y_i} \]Here, \(y_i\) denotes whether the outcome is a success (1) or failure (0).
The Bernoulli distribution is foundational for understanding larger distribution models, such as the binomial distribution, which involves multiple Bernoulli trials.
  • In practical terms, a Bernoulli distribution could model scenarios like a coin flip, with \(p\) as the probability of landing heads.
  • It is crucial in forming the basis for maximum likelihood estimation and other statistical inference techniques.
Binomial distribution
The binomial distribution extends the concept of the Bernoulli distribution by modeling the number of successes in \(n\) independent Bernoulli trials, each with a success probability \(p\). It is represented as \(B(n, p)\).
The binomial probability that there are exactly \(k\) successes in \(n\) trials is given by:\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]where \(\binom{n}{k}\) is the binomial coefficient, which counts the number of ways \(k\) successes can occur in \(n\) trials.
This distribution is used widely in hypothesis testing and helps determine critical values like \(k^*\) in test criteria, where the outcome of interest is the sum of successes across trials.
  • Applications for the binomial distribution include quality control and surveys, where you test how often a characteristic arises in a population.
  • It underlies many statistical testing methods and aids in constructing confidence intervals.

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Most popular questions from this chapter

What assumptions are made about the populations from which independent random samples are obtained when the \(t\) distribution is used to make small-sample inferences concerning the differences in population means?

Why is the \(Z\) test usually inappropriate as a test procedure when the sample size is small?

Researchers have shown that cigarette smoking has a deleterious effect on lung function. In their study of the effect of cigarette smoking on the carbon monoxide diffusing capacity (DL) of the lung, Ronald Knudson, W. Kaltenborn and B. Burrows found that current smokers had DL readings significantly lower than either ex-smokers or nonsmokers. \(^{*}\) The carbon monoxide diffusing capacity for a random sample of current smokers was as follows: $$\begin{array}{rrrrr} 103.768 & 88.602 & 73.003 & 123.086 & 91.052 \\ 92.295 & 61.675 & 90.677 & 84.023 & 76.014 \\ 100.615 & 88.017 & 71.210 & 82.115 & 89.222 \\ 102.754 & 108.579 & 73.154 & 106.755 & 90.479 \end{array}$$ Do these data indicate that the mean DL reading for current smokers is lower than 100 , the average DL reading for nonsmokers? a. Test at the \(\alpha=.01\) level. b. Bound the \(p\) -value using a table in the appendix. c. Find the exact \(p\) -value.

Let \(Y_{1}\) and \(Y_{2}\) be independent and identically distributed with a uniform distribution over the interval \((\theta, \theta+1)\). For testing \(H_{0}: \theta=0\) versus \(H_{a}: \theta>0,\) we have two competing tests: Test 1: Reject \(H_{0}\) if \(Y_{1}>.95\). Test 2: Reject \(H_{0}\) if \(Y_{1}+Y_{2}>c\). Find the value of \(c\) so that test 2 has the same value for \(\alpha\) as test \(1 .\) [Hint: In Example 6.3 , we derived the density and distribution function of the sum of two independent random variables that are uniformly distributed on the interval \((0,1) .]\)

An Article in American Demographics investigated consumer habits at the mall. We tend to spend the most money when shopping on weekends, particularly on Sundays between 4:00 and 6:00 P.M. Wednesday-morning shoppers spend the least. \(^{\star}\) Independent random samples of weekend and weekday shoppers were selected and the amount spent per trip to the mall was recorded as shown in the following table: $$\begin{array}{ll} \hline \text { Weekends } & \text { Weekdays } \\ \hline n_{1}=20 & n_{2}=20 \\ \bar{y}_{1}=\$ 78 & \bar{y}_{2}=\$ 67 \\ s_{1}=\$ 22 & s_{2}=\$ 20 \\ \hline \end{array}$$ a. Is there sufficient evidence to claim that there is a difference in the average amount spent per trip on weekends and weekdays? Use \(\alpha=.05\) b. What is the attained significance level?
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