91Ó°ÊÓ

Sometimes experiments involving success or failure responses are run in a paired or before/after manner. Suppose that before a major policy speech by a political candidate, \(n\) individuals are selected and asked whether \((S)\) or not \((F)\) they favor the candidate. Then after the speech the same \(n\) people are asked the same question. The responses can be entered in a table as follows: After \(S \quad F\) where \(x_{1}+x_{2}+x_{3}+x_{4}=n\). Let \(p_{1}, p_{2}, p_{3}\), and \(p_{4}\) denote the four cell probabilities, so that \(p_{1}=P(S\) before and \(S\) after), and so on. We wish to test the hypothesis that the true proportion of supporters \((S)\) after the speech has not increased against the alternative that it has increased. a. State the two hypotheses of interest in terms of \(p_{1}, p_{2}, p_{3}\), and \(p_{4}\). b. Construct an estimator for the after/before difference in success probabilities. c. When \(n\) is large, it can be shown that the rv \(\left(X_{f}-X_{j}\right) / n\) has approximately a normal distribution with variance given by \(\left[p_{i}+p_{j}-\left(p_{j}-p_{j}\right)^{2}\right] / n\). Use this to construct a test statistic with approximately a standard normal distribution when \(H_{0}\) is true (the result is called McNemar's test). d. If \(x_{1}=350, \quad x_{2}=150, \quad x_{3}=200\), and \(x_{4}=300\), what do you conclude?

Step by step solution

01

State the Hypotheses

We need to consider the probabilities for before and after responses: - Let \(p_{1} = P(S \text{ before and } S \text{ after})\),- \(p_{2} = P(F \text{ before and } S \text{ after})\),- \(p_{3} = P(S \text{ before and } F \text{ after})\),- \(p_{4} = P(F \text{ before and } F \text{ after})\).The null hypothesis \( H_0 \) is that the proportion of supporters has not increased: \( p_1 + p_2 \leq p_1 + p_3 \).The alternative hypothesis \( H_a \) is that the proportion of supporters has increased: \( p_1 + p_2 > p_1 + p_3 \), which simplifies to \( p_2 > p_3 \).

02

Construct Estimator for Success Probability Difference

We want to estimate the difference in success probabilities, which is the difference in proportions after and before the speech:\[\hat{d} = \frac{x_2 + x_1}{n} - \frac{x_3 + x_1}{n} = \frac{x_2 - x_3}{n}\]This estimator represents the difference in the proportion of individuals who supported the candidate after and before the speech, ignoring the constant support row.

03

Construct Test Statistic (McNemar's Test)

To test the hypothesis, we use the given variance to form a test statistic with a standard normal distribution. The test statistic for McNemar's test is:\[Z = \frac{x_2 - x_3}{\sqrt{x_2 + x_3}}\]Under the null hypothesis, this statistic is approximately normally distributed with mean 0 and standard variance 1.

04

Calculate Test Statistic with Given Data

Given \(x_1 = 350\), \(x_2 = 150\), \(x_3 = 200\), and \(x_4 = 300\), we calculate the test statistic using the formula:\[Z = \frac{150 - 200}{\sqrt{150 + 200}} = \frac{-50}{\sqrt{350}} \approx \frac{-50}{18.71} \approx -2.67\]

05

Decision and Conclusion

The calculated test statistic \(Z \approx -2.67\) is compared to a critical value from the standard normal distribution, typically \(\pm 1.96\) for a 5% significance level. Since \(-2.67 < -1.96\), it falls beyond the critical region, leading us to reject the null hypothesis. This suggests that the proportion of supporters for the candidate actually decreased rather than increased after the speech.

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Key Concepts

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

Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics used to make decisions based on data. It involves formulating two opposing hypotheses and using data to determine which one is more plausible.

• The **null hypothesis** (\( H_0 \)): This is a statement that indicates no effect or no change in the population parameter. In our case, it suggests that the proportion of supporters after the speech has not increased.
• The **alternative hypothesis** (\( H_a \)): This hypothesis contradicts the null and is what the researcher aims to prove. Here, it claims that the proportion of supporters has increased.

We make decisions by comparing the calculated test statistics against a critical value, derived from a probability distribution, such as the standard normal distribution. If the test statistic is in the critical region, we reject the null hypothesis.

Success Probability

Success probability pertains to the likelihood of a favorable outcome in a trial. In this exercise, it refers to the probability that individuals will support the candidate before and after a speech. These probabilities can be denoted as:

• \( p_1 \): Probability of supporting both before and after.
• \( p_2 \): Probability of opposing before but supporting after.
• \( p_3 \): Probability of supporting before but opposing after.
• \( p_4 \): Probability of opposing both before and after.

Estimating these probabilities provides insights into change in support.

The difference in these probabilities, notably \( p_2 - p_3 \), forms the basis to determine if the speech had a positive impact on the number of supporters.

Standard Normal Distribution

The standard normal distribution is a key tool in hypothesis testing. It is a normal distribution with a mean of 0 and a standard deviation of 1. Many test statistics, including those in McNemar's test, are designed to follow this distribution under the null hypothesis.

For McNemar’s test, the test statistic \( Z \) is calculated as \( \frac{x_2 - x_3}{\sqrt{x_2 + x_3}} \). Under the null hypothesis, \( Z \) follows a standard normal distribution, which allows us to determine the critical values.

• A test statistic lying within critical values (e.g., at \( \pm 1.96 \) for a 5% significance level) supports the null hypothesis.
• Falling outside these bounds suggests rejection of the null, indicating statistical significance.

Experimental Design

Experimental design provides structure to experiments to ensure valid and reliable results. In paired experiments, like the before-and-after study about a political speech, each subject acts as their own control.

Such designs help in:

• **Reducing variability:** By comparing each individual against their own baseline, external influences are minimized.
• **Increasing accuracy:** This method emphasizes changes due to the intervention (e.g., the speech) rather than individual differences.

Through careful design and analysis, researchers ensure that observed changes are due to the experiment, not extraneous factors.