91Ó°ÊÓ

In a survey of 3029 adult Americans, the Harris Poll asked people whether they smoked cigarettes and whether they always wear a seat belt in a car. The table shows the results of the survey. For each activity, we define a success as finding an individual who participates in the hazardous activity. $$ \begin{array}{lcc} & \begin{array}{c} \text { No Seat Belt } \\ \text { (success) } \end{array} & \begin{array}{c} \text { Seat Belt } \\ \text { (failure) } \end{array} \\ \hline \text { Smoke (success) } & 67 & 448 \\ \hline \text { Do not smoke (failure) } & 327 & 2187 \\ \hline \end{array} $$ (a) Why is this a dependent sample? (b) Is there a significant difference in the proportion of individuals who smoke and the proportion of individuals who do not wear a seat belt? In other words, is there a significant difference between the proportion of individuals who engage in hazardous activities? Use the \(\alpha=0.05\) level of significance.

Step by step solution

01

Identify the type of sample (Part a)

This is a dependent sample because the same group of individuals was surveyed for both activities (smoking and wearing a seat belt).

02

Define hypotheses (Part b)

Set up the null and alternative hypotheses. Null hypothesis (H_0): There is no significant difference in the proportion of individuals who smoke and those who do not wear a seat belt. Alternative hypothesis (H_a): There is a significant difference between the proportions. Formally: \[ H_0: p_1 - p_2 = 0 \] \[ H_a: p_1 - p_2 eq 0 \]

03

Calculate sample proportions

Calculate the sample proportions for each hazardous activity. \( p_1 \) (proportion of smokers) = \( \frac{67 + 448}{3029} \) = \( \frac{515}{3029} \) \( p_2 \) (proportion of non-seat belt users) = \( \frac{67 + 327}{3029} \) = \( \frac{394}{3029} \)

04

Find pooled proportion

Calculate the pooled proportion \( p \) using the formula: \[ p = \frac{x_1 + x_2}{n_1 + n_2} = \frac{515 + 394}{3029 + 3029} = \frac{909}{6058} \]

05

Compute the standard error

Calculate the standard error (SE) using the formula: \[ SE = \sqrt{p(1-p) \left( \frac{1}{n_1}+ \frac{1}{n_2} \right) } = \sqrt{ \frac{909}{6058} \cdot \left(1 - \frac{909}{6058} \right) \cdot \left( \frac{1}{3029} + \frac{1}{3029} \right)} \]

06

Compute test statistic

Calculate the test statistic (z): \[ z = \frac{(p_1 - p_2)}{SE} \]

07

Find critical value and make a decision

Compare the test statistic to the critical value at \( \alpha = 0.05 \) for a two-tailed test, which is \( z_{\alpha/2} = 1.96 \). If \( |z| > 1.96 \), reject the null hypothesis. If \( |z| \leq 1.96 \), fail to reject the null hypothesis.

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Key Concepts

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

Dependent Sample

A dependent sample, also called a paired sample, occurs when the observations in one sample are related to the observations in another sample. This relationship might be a one-to-one correspondence, such as two measurements on the same individual. In the given exercise, the same group of 3029 adult Americans was surveyed for two traits: smoking habits and seat belt usage. Since these observations come from the same group of individuals, we consider the sample to be dependent. This dependency is crucial in the analysis because it means the traits (smoking and seat belt usage) might influence each other, adding depth to our statistical investigation.

Proportion Hypothesis Test

A proportion hypothesis test is a statistical method used to determine whether there is a significant difference between two population proportions. In our case, we want to test if the proportion of people who smoke (a hazardous activity) differs significantly from the proportion of people who do not wear a seat belt (another hazardous activity). To do this, we set up two hypotheses:

• The null hypothesis (\(H_0\)) states that there is no difference in proportions: \(p_1 - p_2 = 0\).
• The alternative hypothesis (\(H_a\)) suggests that there is a difference: \(p_1 - p_2 eq 0\).

By analyzing sample data, we use statistical calculations to decide whether the observed differences are likely due to random chance or if they are statistically significant.

Pooled Proportion

The pooled proportion is used when performing a hypothesis test for the difference between two proportions. It combines the success counts from both samples into a single proportion to estimate a common probability of success across both groups. In our exercise, we calculate the pooled proportion (\( p \)) as follows:
\[ p = \frac{x_1 + x_2}{n_1 + n_2} \]
Here, \( x_1\) and \( x_2\) are the number of successes (individuals who smoke and those who do not wear a seat belt, respectively). We sum these and divide by the total number of observations, giving us a single estimate for the population proportion. This pooled value is then used to calculate the standard error and, eventually, the test statistic.

Standard Error

The standard error (SE) measures the variability of a sample statistic. For the proportion hypothesis test, it assesses the spread of the difference between the sample proportions. We compute the SE using the pooled proportion (\( p \)) and the sample sizes (\( n_1\) and \( n_2\)). The formula is:
\[ SE = \sqrt{p(1-p)\left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \]
This equation incorporates the combined variance of the two sample proportions, giving us a sense of how much we can expect the sampled difference to fluctuate due to random sampling. A smaller SE indicates more precise estimates, which strengthens the reliability of our test results.

Z-Test

The Z-test is a statistical test used to determine whether there is a significant difference between sample and population proportions. In this exercise, we compare the calculated Z value to a critical value from the Z-distribution. Here's how we compute the test statistic (\( z \)):
\[ z = \frac{(p_1 - p_2)}{SE} \]
Where \( p_1 \) and \( p_2 \) are the sample proportions, and SE is the standard error calculated earlier. We then compare the calculated Z value to the critical values at a chosen level of significance (\( \alpha = 0.05\)). If the absolute Z value exceeds the critical value (1.96 for a two-tailed test), we reject the null hypothesis, indicating a significant difference between the proportions. Conversely, if the Z value falls within the critical range, we fail to reject the null hypothesis.