91Ó°ÊÓ

Full body scan, Part I. A news article reports that "Americans have differing views on two potentially inconvenient and invasive practices that airports could implement to uncover potential terrorist attacks." This news piece was based on a survey conducted among a random sample of 1,137 adults nationwide, interviewed by telephone November \(7-10,2010,\) where one of the questions on the survey was "Some airports are now using 'full-body' digital x-ray machines to clectronically screen passengers in airport security lines. Do you think these new x-ray machines should or should not be used at airports?" Below is a summary of responses based on party affiliation. \begin{tabular}{llccc} & & \multicolumn{3}{c} { Party Affilation } \\ \cline { 3 - 5 } & & Republican & Democrat & Independent \\ \cline { 2 - 5 } Answer & Should & 264 & 299 & 351 \\ & Should not & 38 & 55 & 77 \\ & Don't know/No answer & 16 & 15 & 22 \\ \cline { 2 - 5 } & Total & 318 & 369 & 450 \end{tabular} (a) Conduct an appropriate hypothesis test evaluating whether there is a difference in the proportion of Republicans and Democrats who think the full- body scans should be applied in airports. Assume that all relevant conditions are met. (b) The conclusion of the test in part (a) may be incorrect, meaning a testing error was made. If an error was made, was it a Type I or a Type II error? Explain.

Step by step solution

01

Define the Hypotheses

To test whether there is a difference in the proportion of Republicans and Democrats who think full-body scans should be used, we set up our null and alternative hypotheses. Let \( p_R \) be the proportion of Republicans and \( p_D \) the proportion of Democrats who think scans should be used. Null Hypothesis \( (H_0) \): \( p_R = p_D \)Alternative Hypothesis \( (H_a) \): \( p_R eq p_D \)

02

Calculate Sample Proportions

Using the data, calculate the sample proportions for Republicans and Democrats.For Republicans, \( \hat{p}_R = \frac{264}{318} \approx 0.830 \).For Democrats, \( \hat{p}_D = \frac{299}{369} \approx 0.810 \).

03

Calculate the Pooled Proportion

Since we are assuming the null hypothesis is true, we calculate the pooled proportion \( \hat{p} \) from both groups:\[ \hat{p} = \frac{264 + 299}{318 + 369} = \frac{563}{687} \approx 0.820 \].

04

Compute the Standard Error

The standard error for the difference in sample proportions is given by:\[ SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_R} + \frac{1}{n_D} \right)} \] where \( n_R = 318 \) and \( n_D = 369 \).\[ SE = \sqrt{0.820 \times 0.180 \times \left( \frac{1}{318} + \frac{1}{369} \right)} \approx 0.033 \].

05

Calculate the Test Statistic

Calculate the test statistic for the difference in proportions using:\[ z = \frac{\hat{p}_R - \hat{p}_D}{SE} = \frac{0.830 - 0.810}{0.033} \approx 0.606 \].

06

Determine the p-value

Using the z-value, look up the corresponding p-value in the standard normal distribution table. For a two-tailed test, this z-value \( 0.606 \) corresponds to a p-value of about 0.544.

07

Conclusion of Hypothesis Test

Compare the calculated p-value with the significance level (commonly 0.05). Since 0.544 > 0.05, we fail to reject the null hypothesis. There is no statistically significant difference in the proportions who think the scans should be used between Republicans and Democrats.

08

Identify the Potential Error

Since we failed to reject the null hypothesis when we do not have evidence for a difference, if this decision is incorrect, we would have a Type II error. A Type II error occurs when we fail to reject a false null hypothesis.

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.

Proportions

Proportions play a significant role in statistics, especially when comparing different groups. In hypothesis testing, a proportion represents a part of the whole, providing valuable insight into group opinions or behaviors. For example, calculating proportions helps us understand what share of Republicans or Democrats believe full-body scans should be used at airports.
Proportions are calculated by dividing the number of favorable outcomes by the total number of observations in the group. Here, for Republicans, the proportion is computed by dividing the number of Republicans in favor of scans (264) by the total surveyed Republicans (318):

• For Republicans, \( \hat{p}_R = \frac{264}{318} \approx 0.830 \).
• For Democrats, \( \hat{p}_D = \frac{299}{369} \approx 0.810 \).

These proportions form the basis for comparing if there's a significant difference in opinions across political lines.

Null Hypothesis

The null hypothesis, denoted as \( H_0 \), is a fundamental part of hypothesis testing. It serves as the default assumption about a population parameter, suggesting that there is no effect or no difference between groups. In this case, the null hypothesis states there is no difference in the proportion of Republicans and Democrats who support full-body scans.
The goal of hypothesis testing is often to determine whether there is enough evidence to reject the null hypothesis. Here, we define:

• Null Hypothesis \( (H_0) \): \( p_R = p_D \)

Where \( p_R \) and \( p_D \) are the proportions of Republicans and Democrats, respectively, in favor of using x-ray machines.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_a \), offers a contrasting viewpoint to the null hypothesis. It suggests there is an effect, or there is a difference between groups—a hypothesis that reflects researchers' interests. Here, our alternative hypothesis is that there is a difference between the proportions of Republicans and Democrats who support using full-body scans.
In this exercise, the alternative hypothesis is formulated as:

• Alternative Hypothesis \( (H_a) \): \( p_R eq p_D \)

This statement indicates we are interested in finding whether the proportions are not equal, exploring any significant variation between the two groups' opinions.

Standard Error

Standard error (SE) measures the variability of a sample statistic as an estimator for a population parameter. It's crucial in hypothesis testing to understand the precision of sample estimates, particularly when comparing proportions. The standard error indicates how much the sample proportion is expected to fluctuate from the actual population proportion.
For the difference in proportions between Republicans and Democrats, the standard error is calculated as:\[SE = \sqrt{\hat{p}(1-\hat{p}) \left( \frac{1}{n_R} + \frac{1}{n_D} \right)}\]where \( \hat{p} \) is the pooled proportion, and \( n_R \) and \( n_D \) are the sample sizes for Republicans and Democrats respectively. Using the given data, the SE is approximately 0.033, providing a measure of how much the estimated difference in proportions may vary by sampling error.

Type I Error

A Type I error arises when the null hypothesis is wrongly rejected, essentially a false positive finding. In hypothesis testing, this means concluding that there is an effect or a difference when, in fact, there isn’t one.
This error is influenced by the significance level. If our significance level is set to 0.05, a Type I error would occur 5% of the time when the null hypothesis is actually true. In the context of the problem, a Type I error would incorrectly suggest that there's a significant difference in opinions between Republicans and Democrats, even when they actually agree.

Type II Error

A Type II error occurs when we fail to reject a null hypothesis that is actually false, meaning a false negative result. This error results in missing a potential finding where a true difference exists but goes undetected.
In the exercise provided, failing to reject the null hypothesis when Republicans and Democrats truly have different opinions on full-body scans represents a Type II error. This error relates directly to the test's power: the higher the study's power, the less likely to commit a Type II error. Unfortunately, more rigorous samples or significant true differences are often needed to minimize Type II errors, meaning statistical tests need careful preparation and execution.

Significance Level

The significance level, usually denoted by \( \alpha \), sets the threshold for deciding whether to reject the null hypothesis. It dictates the probability of making a Type I error, where a common significance level is 0.05. This means a researcher allows a 5% chance of incorrect rejection of the null hypothesis.
In hypothesis testing, if the p-value calculated from the test is less than or equal to the significance level, the null hypothesis is rejected. In this exercise, the p-value was 0.544, which is greater than the typical 0.05 significance level, leading us to fail to reject the null hypothesis. Understanding and setting the significance level is crucial because it balances the risk of errors against practical research considerations.