91Ó°ÊÓ

The Insurance Institute for Highway Safety issued a news release titled "Teen Drivers Often Ignoring Bans on Using Cell Phones" (June 9,2008 ). The following quote is from the news release: Just \(1-2\) months prior to the ban's Dec. \(1,2006,\) start, \(11 \%\) of teen drivers were observed using cell phones as they left school in the afternoon. About 5 months after the ban took effect, \(12 \%\) of teen drivers were observed using cell phones. Suppose that the two samples of teen drivers (before the ban, after the ban) are representative of these populations of teen drivers. Suppose also that 200 teen drivers were observed before the ban (so \(n_{1}=200\) and \(\hat{p}_{1}=0.11\) ) and that 150 teen drivers were observed after the ban. a. Construct and interpret a \(95 \%\) large-sample confidence interval for the difference in the proportion using a cell phone while driving before the ban and the proportion after the ban. b. Is 0 included in the confidence interval of Part (a)? What does this imply about the difference in the population proportions?

Step by step solution

01

Identify the sample proportions and sample sizes.

Before the ban: \(n_1 = 200\), \(\hat{p}_1 = 0.11\) After the ban: \(n_2 = 150\), \(\hat{p}_2 = 0.12\)

02

Calculate the combined proportion.

We will calculate the combined proportion, \(\hat{P}\), by using the equation: \(\hat{P} = \frac{n_1\hat{p}_1 + n_2\hat{p}_2}{n_1 + n_2}\) \(\hat{P} = \frac{(200)(0.11) + (150)(0.12)}{200 + 150}\) \(\hat{P} = \frac{22 + 18}{350}\) \(\hat{P} = \frac{40}{350}\) \(\hat{P} = 0.11429\)

03

Calculate the standard error.

Now, we will calculate the standard error of the difference between the two proportions using the equation: \(SE = \sqrt{\frac{\hat{P}(1 - \hat{P})}{n_1} + \frac{\hat{P}(1 - \hat{P})}{n_2}}\) \(SE = \sqrt{\frac{(0.11429)(1 - 0.11429)}{200} + \frac{(0.11429)(1 - 0.11429)}{150}}\) \(SE = \sqrt{\frac{0.10128175}{200} + \frac{0.10128175}{150}}\) \(SE = \sqrt{0.00050641 + 0.00067521}\) \(SE = 0.038714\)

04

Calculate the 95% confidence interval.

In order to construct a 95% confidence interval for the difference in proportions, we'll use a z-value of 1.96 (rounded to 2 for simplicity). The confidence interval is given by the equation: \((\hat{p}_1 - \hat{p}_2) \pm z \times SE\) \((0.11 - 0.12) \pm 2 \times 0.038714\) \((-0.01) \pm 0.077428\) The 95% confidence interval is (-0.087428, 0.067428). #b. Is 0 included in the confidence interval of Part (a)? What does this imply about the difference in the population proportions?# As we can see, 0 is included in the confidence interval (-0.087428, 0.067428). This means that we cannot reject the null hypothesis that there is no difference in the proportion of teen drivers using cell phones while driving before and after the ban. In other words, the ban does not seem to have a significant effect on the proportion of teen drivers using cell phones while driving.

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.

Statistics Education

The field of statistics plays a pivotal role in understanding data and making data-driven decisions across various domains. One fundamental topic within statistics education is the concept of a confidence interval, which provides a range of values that likely contain the true population parameter.

For instance, in our exercise, the confidence interval is used to understand the difference in proportions of teen drivers using cell phones before and after a ban. The education in statistics equips students not only with the skills to calculate such intervals but also with the critical thinking abilities to interpret the results, which is crucial for real-world applications.

Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences about a population, based on sample data. In the context of our example, the null hypothesis might state that the ban on using cell phones had no effect on the behavior of teen drivers.

To test this hypothesis, we use the confidence interval for the difference in proportions. If the interval contains zero (as it does in our exercise), it suggests that the observed difference could be due to random sample variability, and we do not have enough evidence to reject the null hypothesis. Understanding and performing hypothesis testing is a bedrock of statistical analysis, teaching students how to objectively assess claims based on quantitative evidence.

Standard Error Calculation

The standard error measures the variability of a sample statistic from the population parameter. It's vital for constructing confidence intervals and conducting hypothesis tests.

In our exercise, the standard error calculation for the difference in sample proportions is critical to determine the confidence interval's width. It involves combining the variance from both populations to capture the total error in the estimate of the difference. A smaller standard error usually leads to a narrower confidence interval, suggesting a more precise estimate. This calculation is a cornerstone of inferential statistics, helping students grasp the reliability of their estimates.

Sample Proportions

Sample proportions represent the fraction of the sample that exhibits a characteristic of interest. In our example, they are the observed percentages of teen drivers using cell phones before and after the ban.

Understanding sample proportions is essential in statistics education because they serve as estimators for true population proportions. The exercise demonstrates how sample proportions are used to estimate the effect of the cell phone ban on teen drivers' behavior, highlighting the importance of representative sampling and providing students with practical insights into assessing proportions in populations.