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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: 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 \(\left(X_{i}-X_{j}\right) / n\) has approximately a normal distribution with variance \(\left[p_{i}+p_{j}-\left(p_{i}-p_{j}\right)^{2}\right] / n\). Use this to construct a test statistic with approximately a standard normal distribution when \(H_{0}\) is true (the result is called MeNemar's test). d. If \(x_{1}=350, x_{2}=150, x_{3}=200\), and \(x_{4}=300\), what do you conclude?

Step by step solution

01

Formulate Hypotheses

To address part (a), we need to state the null and alternative hypotheses. The null hypothesis, \(H_0\), is that the proportion of supporters after the speech has not increased compared to before, which means \(p_2 + p_1 = p_3 + p_1\) or equivalently \(p_2 = p_3\). The alternative hypothesis, \(H_a\), is that the proportion of supporters after has increased, i.e., \(p_2 > p_3\).

02

Construct Estimator for Difference

For part (b), we need to construct an estimator for the difference in success probabilities after versus before the speech. The estimator of interest is \(\hat{p}_2 - \hat{p}_3 = \frac{x_2}{n} - \frac{x_3}{n}\). This represents the difference in proportions between those who switched from support to no support and those who switched from no support to support.

03

Construct Test Statistic

To address part (c), we utilize the given information to construct a test statistic for MeNemar's test. Given large \(n\), \(\frac{(X_2 - X_3)}{n}\) follows an approximate normal distribution. The variance is \(\left[p_2 + p_3 - (p_2 - p_3)^2\right] / n\). Under \(H_0\), the test statistic is \(Z = \frac{X_2 - X_3}{\sqrt{X_2 + X_3}}\), which approximately follows a standard normal distribution when the null hypothesis is true.

04

Compute and Interpret

For part (d), substitute the given values: \(x_1 = 350\), \(x_2 = 150\), \(x_3 = 200\), \(x_4 = 300\). Compute the test statistic \(Z = \frac{150 - 200}{\sqrt{150 + 200}} = \frac{-50}{\sqrt{350}} = \frac{-50}{18.71} \approx -2.67\). Using a standard normal distribution table, \(Z = -2.67\) falls in the critical region for typical \(\alpha = 0.05\). Therefore, we reject the null hypothesis.

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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 method used in statistics to determine if there is enough evidence in a sample of data to infer that a particular condition is true for the entire population. It involves making a claim or hypothesis, then testing whether the data supports this claim through statistical analysis.

For the problem at hand, we start with a null hypothesis, denoted as \(H_0\), which assumes no change in the proportion of political candidate supporters before and after a speech. In other words, \(p_2 = p_3\), meaning the number of individuals supporting the candidate before remains consistent after the speech. The alternative hypothesis, \(H_a\), posits that the support has increased (i.e., \(p_2 > p_3\)).

• **Null Hypothesis (\(H_0\))**: \(p_2 = p_3\)
• **Alternative Hypothesis (\(H_a\))**: \(p_2 > p_3\)

Once you've formulated these hypotheses, the next step is performing a statistical test to evaluate them. This involves determining the likelihood of observing data as extreme as the sample data, assuming \(H_0\) is true.

Test Statistic

The test statistic is a standardized value calculated from sample data, used to determine the probability of observing such data under the null hypothesis. It is a central component of hypothesis testing and helps decide whether to reject the null hypothesis.

In the case of McNemar's test, the test statistic \(Z\) is computed using the formula:\[Z = \frac{X_2 - X_3}{\sqrt{X_2 + X_3}}\]This formula is specific to situations where paired samples are involved, like comparing supporter responses before and after an event. Here, \(X_2\) and \(X_3\) represent counts of changed opinions – those who supported before but not after, and vice versa. The numerator \(X_2 - X_3\) is the observed difference in paired proportions, while the denominator \(\sqrt{X_2 + X_3}\) acts as a scaling factor to account for variability.

• **Purpose**: To determine if the observed difference is statistically significant.
• **Significance**: Values which fall far from zero can indicate strong evidence against the null hypothesis.

Through this calculation, one can assess whether any changes in opinion are significant enough to reject the null hypothesis in favor of the alternative.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve, mean, and variance. It is heavily utilized in statistics due to its properties lining up with many natural phenomena, such as measurement errors and human characteristics.

For McNemar's test, the large sample size allows us to approximate the distribution of the test statistic to the normal distribution. This approximation is crucial because it enables the use of tables for the standard normal distribution in making statistical inferences. By assuming a normal distribution for the test statistic under the null hypothesis, we can calculate the probability of observing a test statistic at least as extreme as the one calculated.

• **Standard Normal Distribution**: A specific kind of normal distribution with a mean of 0 and a standard deviation of 1.
• **Role in McNemar's Test**: Facilitates determining critical values and p-values for hypothesis testing.

Using the normal distribution, we can meaningfully interpret the test statistic calculated in the earlier step, by seeing how it compares to the expected standard normal distribution given \(H_0\).

Statistical Inference

Statistical inference involves using data from a sample to make conclusions about a larger population. It combines data analysis with probability to predict and interpret real-world phenomena.

In this context, McNemar's test is a method of conducting statistical inference about proportions of binary paired data. It allows us to infer whether there has been a statistically significant change in the proportion of people who support the candidate before and after the speech. This is done by calculating the test statistic and comparing it to known statistical distributions to make informed decisions about the null hypothesis.

• **Objective**: To identify whether observed changes in sample data reflect real changes in the population.
• **Confidence Level and Significance Level**: These determine the reliability of inferences. A common significance level is \(\alpha = 0.05\), meaning there's a 5% risk of concluding a difference exists when there is none.

Ultimately, statistical inference through tests like McNemar's provides a robust framework for understanding and validating changes in paired data scenarios, helping us interpret whether observed differences are likely to be true in the broader context.