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Chapter 16: Problem 21

Applet Exercise In Exercise 16.15, we determined that the posterior density for \(p\), the proportion of responders to the new treatment for a virulent disease, is a beta density with parameters \(\alpha^{*}=5\) and \(\beta^{*}=24 .\) What is the conclusion of a Bayesian test for \(H_{0}: p<.3\) versus \(H_{a}:\) \(p \geq .3 ?\) [Use the applet Beta Probabilities and Quantiles at https://college.cengage.com/nextbook/statistics/wackerly 966371/student/html/index.html. Alternatively, if \(W\) is a beta-distributed random variable with parameters \(\alpha\) and \(\beta\), the \(R\) or \(S\) -Plus command pbeta \((w, \alpha, \beta) \text { gives } P(W \leq w) .]\)

Short Answer

Expert verified
Calculate \(P(p < 0.3)\) using the `pbeta` function; reject \(H_0\) if this probability is significantly low.

Step by step solution

01

Understand the Beta Distribution

The exercise provides that the posterior density for the proportion of responders, denoted as \(p\), follows a Beta distribution with parameters \(\alpha^* = 5\) and \(\beta^* = 24\). A Beta distribution, \(Beta(\alpha, \beta)\), is used to model probabilities and is defined for \(0 \leq p \leq 1\).
02

Define Hypotheses

The null hypothesis (\(H_0\)) is that the proportion \(p\) is less than 0.3, and the alternative hypothesis (\(H_a\)) is that \(p\) is 0.3 or greater. Mathematically, \(H_0: p < 0.3\) versus \(H_a: p \geq 0.3\).
03

Determine Test Statistic

Using the condition \(H_0: p < 0.3\), we need to calculate \(P(p < 0.3)\) using the specified Beta distribution. This evaluates probability for the null hypothesis.
04

Calculate Probability Using R/Python's pbeta Function

To find \(P(p < 0.3)\), you can utilize the `pbeta` function. This is done by executing `pbeta(0.3, 5, 24)` in R or through a similar computational method in Python. This function calculates the cumulative distribution function value for \(p < 0.3\).
05

Interpret Results

The result from the `pbeta` function provides the probability that a Beta(5, 24)-distributed random variable is less than 0.3. If this probability, \(P(p < 0.3)\), is low, we reject the null hypothesis in favor of the alternative that \(p \geq 0.3\). Otherwise, we do not reject \(H_0\).
06

Conclusion

If \(P(p < 0.3) < 0.05\), the result is statistically significant, and we reject \(H_0\), concluding there is evidence that \(p \geq 0.3\). If \(P(p < 0.3)\) is greater than 0.05, we do not reject \(H_0\), suggesting insufficient evidence to conclude \(p \geq 0.3\).

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Key Concepts

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

Beta Distribution
The Beta distribution is a key tool used in Bayesian statistics. It is particularly helpful in modeling probabilities over bounded intervals, like the interval from 0 to 1 usually for proportion data. The distribution is defined by two parameters, \(\alpha\) and \(\beta\), which shape the curve of the distribution, allowing flexibility to fit different kinds of data.
  • The Beta distribution is often used when there is a need to model and estimate the probability of a certain event occurring.
  • The parameters \(\alpha\) and \(\beta\) directly influence the mean and variance of the data being modeled.
  • In the context of the given problem, \(\alpha^*=5\) and \(\beta^*=24\), meaning the distribution is used to understand the density of responders to a treatment.
Understanding how these parameters shape the probability distribution helps in interpreting the probability and making informed statistical conclusions.
Hypothesis Testing
Hypothesis testing is a method of making statistical decisions using data. In Bayesian statistics, it's used to assess hypotheses about parameters based on observed data. The goal is to determine whether there is enough evidence in the data to support a certain hypothesis.
  • A null hypothesis (\(H_0\)) is a statement of no effect or no difference. It is usually what you aim to challenge or test.
  • The alternative hypothesis (\(H_a\)) is what you consider if the null hypothesis is rejected, suggesting there is an effect or a difference.
In this exercise, we test whether the proportion of responders, \(p\), is either less than 0.3 (\(H_0: p < 0.3\)) or at least 0.3 (\(H_a: p \geq 0.3\)). The decision to reject or not reject \(H_0\) is based on the probability derived from the posterior distribution, which is a Beta distribution in this case.
Posterior Density
In Bayesian statistics, posterior density is central as it represents what you know about a parameter after considering both prior beliefs and new data. It embodies the updated belief about the parameter after observing data.
  • The posterior density is obtained by combining prior distribution and likelihood from data using Bayes' theorem.
  • It provides a complete possible range of parameter values and their corresponding probabilities.
  • In this context, the posterior density helps decide which hypothesis to support by using the calculated probabilities for the Beta-distributed random variable.
The calculated posterior density for \(p < 0.3\) informs whether to reject the null hypothesis. If this probability is low, it indicates that there is enough evidence against \(H_0\), suggesting support for \(H_a: p \geq 0.3\).

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Most popular questions from this chapter

In Exercise 16.7 , we reconsidered our introductory example where the number of responders to a treatment for a virulent disease in a sample of size \(n\) had a binomial distribution with parameter \(p\) and used a beta prior for \(p\) with parameters \(\alpha=1\) and \(\beta=3 .\) We subsequently found that, upon observing \(Y=y\) responders, the posterior density function for \(p | y\) is a beta density with parameters \(\alpha^{*}=y+\alpha=y+1\) and \(\beta^{*}=n-y+\beta=n-y+3 .\) If we obtained a sample of size \(n=25\) that contained 4 people who responded to the new treatment, find a \(95 \%\) credible interval for \(p\). [Use the applet Beta Probabilities and Quantiles at Alternatively, if \(W\) is a beta-distributed random variable with parameters \(\alpha\) and \(\beta\), the \(R\) (or \(S\) -Plus) command qbeta \((\mathrm{p}, \alpha, \beta) \text { gives the value } w \text { such that } P(W \leq w)=p .]\)

In Section 16.1 and Exercise 16.6 , we considered an example where the number of responders to a treatment for a virulent disease in a sample of size \(n\) had a binomial distribution with parameter \(p\) and used a beta prior for \(p\) with parameters \(\alpha=1\) and \(\beta=3.\) a. Find the Bayes estimator for \(p=\) the proportion of those with the virulent disease who respond to the therapy. b. Derive the mean and variance of the Bayes estimator found in part (a).

Suppose that \(Y\) is a binomial random variable based on \(n\) trials and success probability \(p\) (this is the case for the virulent-disease example in Section 16.1 ). Use the conjugate beta prior with parameters \(\alpha\) and \(\beta\) to derive the posterior distribution of \(p | y\). Compare this posterior with that found in Example 16.1.

In Exercise 16.11, we found the posterior density for \(\lambda\) the mean of a Poisson-distributed population. Assuming a sample of size \(n\) and a conjugate gamma \((\alpha, \beta)\) prior for \(\lambda\), we showed that the posterior density of \(\lambda | \sum y_{i}\) is gamma with parameters \(\alpha^{*}=\Sigma y_{i}+\alpha\) and \(\beta^{*}=\beta /(n \beta+1)\). If a sample of size \(n=25\) is such that \(\sum y_{i}=174\) and the prior parameters were \((\alpha=2, \beta=3),\) use the applet Gamma Probabilities and Quantiles to find a \(95 \%\) credible interval for \(\lambda\).

In Exercise 16.9 , we used a beta prior with parameters \(\alpha\) and \(\beta\) and found the posterior density for the parameter \(p\) associated with a geometric distribution. We determined that the posterior distribution of \(p | y\) is beta with parameters \(\alpha^{*}=\alpha+1\) and \(\beta^{*}=\beta+y-1\) Suppose we used \(\alpha=10\) and \(\beta=5\) in our beta prior and observed the first success on trial 6 Determine an \(80 \%\) credible interval for \(p\).
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