91Ó°ÊÓ

For a random sample of \(n\) individuals taking a licensing exam, let \(X_{i}=1\) if the \(i\) th individual in the sample passes the exam and \(X_{i}=0\) otherwise \((i=1, \ldots, n)\). a. With \(p\) denoting the proportion of all examtakers who pass, show that the most powerful test of \(H_{0}: p=.5\) versus \(H_{\mathrm{a}}: p=.75\) rejects \(H_{0}\) when \(\Sigma x_{i} \geq c\). b. If \(n=20\) and you want \(\alpha \leq .05\) for the test of (a), would you reject \(H_{0}\) if 15 of the 20 individuals in the sample pass the exam? c. What is the power of the test you used in (b) when \(p=.75[\) i.e., what is \(\pi(.75)]\) ? d. Is the test derived in (a) UMP for testing the hypotheses \(H_{0}: p=.5\) versus \(H_{\mathrm{a}}: p>.5\) ? Explain your reasoning. e. Graph the power function \(\pi(p)\) of the test for the hypotheses of (d) when \(n=20\) and \(\alpha \leq .05\). f. Return to the scenario of (a), and suppose the test is based on a sample size of 50 . If the probability of a type II error is approximately \(.025\), what is the approximate significance level of the test (use a normal approximation)?

Step by step solution

01

Understand the Test Framework

The task is to find the most powerful test for the given hypothesis involving a binomial distribution, since we are testing a proportion based scenario where each pass or fail event is a Bernoulli trial.

02

Identify the Test Statistics

For the hypothesis test, calculate the sample proportion \( \hat{p} = \frac{\Sigma X_i}{n} \) where \( \Sigma X_i \) is the number of students passing the exam.

03

Determine the Test Critical Value

To reject \( H_0: p = 0.5 \) versus \( H_a: p = 0.75 \) at significance level \( \alpha \), use a binomial distribution \( B(n, p_0) \) to find the critical value \( c \) such that \( P(\Sigma X_i \geq c | H_0) \leq \alpha \).

04

Calculate Critical Value for n=20 and α=0.05

With \( n = 20 \) and \( \alpha = 0.05 \), find the smallest \( c \) such that \( P(\Sigma X_i \geq c | p = 0.5) \leq 0.05 \) using binomial distribution tables or software. This involves looking at the cumulative probabilities.

05

Evaluate Scenario b with Correct Calculations

From binomial distribution tables for \( n = 20 \) and \( p = 0.5 \), determine if \( c = 15 \) results in \( P(\Sigma X_i \geq 15) \leq 0.05 \). This involves checking if \(\Sigma X_i \) calculated at 15 is in the rejection region established by \(c\).

06

Calculate Power of the Test with p=0.75

Power of the test \( \pi(p=0.75) \) is calculated using \( P(\Sigma X_i \geq c | p = 0.75) \). This involves recognizing the alternate hypothesis probability, where a similar cumulative approach is used with the new success probability.

07

Evaluate the UMP for p > 0.5

Uniformly Most Powerful (UMP) test against \( H_a: p > 0.5 \) is not achievable here since the test is designed specifically for \( p = 0.75 \) and is not a one-sided test. Thus, the most powerful test over all \( p > 0.5 \) is not identified.

08

Plot the Power Function π(p) for n=20

Develop a plot by evaluating \( P(\Sigma X_i \geq c | p) \) for different \( p \) values. This results in a power function curve specifying power at various probability levels.

09

Analyze Type II Error and Normal Approximation in Larger Sample

For \( n=50 \) with Type II error \( \beta = 0.025 \), use normal approximation \( \Sigma X_i \sim N(np, np(1-p)) \). Calculate \( \alpha \) from \( 1 - \beta \) region associated with a corresponding power test, comparing this to binomial distribution threshold.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Binomial Distribution

The binomial distribution is a foundational concept in statistics, particularly useful in hypothesis testing involving proportions. When we say a situation follows a binomial distribution, we're referring to an experiment that meets these requirements:

• There are a fixed number of trials, denoted as \( n \).
• Each trial has two possible outcomes: success (e.g., pass) or failure (e.g., fail).
• The probability of success is consistent across all trials, given by \( p \).
• The trials are independent, meaning the outcome of one trial doesn't affect another.

In our exercise, each student's exam outcome is modeled as a Bernoulli trial—pass or fail—with \( X_i = 1 \) if they pass and \( X_i = 0 \) if they fail. Aggregating these outcomes across all students results in a binomial distribution \( B(n, p) \), where \( \sum X_i \) captures the number of students who pass among the \( n \) trials conducted.

Significance Level

The significance level, denoted as \( \alpha \), is crucial when conducting hypothesis tests. It represents the threshold at which we decide whether or not to reject the null hypothesis \( H_{0} \). If the calculated probability of observing the test results (or more extreme) under \( H_{0} \) is less than or equal to \( \alpha \), we reject \( H_{0} \).
For example, in the exercise, the significance level is set to \( \alpha = 0.05 \). This means there is a 5% risk of incorrectly rejecting \( H_{0} \) when it is true (a Type I error). The choice of \( \alpha \) balances sensitivity and specificity in hypothesis testing, impacting the critical value \( c \) beyond which we reject \( H_{0} \). In simpler terms, it helps set the cut-off point for deciding if our sample results are statistically significant.

Type II Error

A Type II error, represented by \( \beta \), is made when the test fails to reject a false null hypothesis. In the exercise, we're concerned with the likelihood of such an error when the true proportion \( p = 0.75 \) under the alternative hypothesis \( H_{a} \), but we mistakenly accept \( H_{0} \). While \( \beta \) indicates the test's blind spots, it also demonstrates a trial's robustness because a low \( \beta \) reduces the risk of overlooking a true effect.
The relationship between \( \beta \) and the power of the test is crucial. As seen in the problem, when the Type II error probability is low, such as \( \beta = 0.025 \), this indicates a high probability of correctly rejecting \( H_{0} \) when it is false, thus improving the test's reliability.

Power of a Test

The power of a test is a statistical measure representing the probability of correctly rejecting a false null hypothesis \( H_{0} \). It is given by \( 1 - \beta \). A powerful test is highly effective at detecting a significant effect when there is one.
In our exercise, the power is particularly relevant when addressing the alternative hypothesis \( H_{a}: p = 0.75 \). The ability to reject \( H_{0} \) accurately when \( p = 0.75 \) reflects our test's robustness. Power increases with larger sample sizes because they provide more information, reducing the variability of results and making deviations from \( H_{0} \) more detectable.
Visualizing the power through a graph, as described in the exercise, aids in understanding how the test's effectiveness varies across different possible values of \( p \). A higher power implies a lower likelihood of making a Type II error, thus increasing the confidence in the test results.