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The Insurance Institute for Highway Safety issucd a press release titled "Teen Drivers Often lgnoring Bans on Using Cell Phones" (June 9,2008 ). The following quote is from the press release: Just \(1-2\) months prior to the ban's \(\underline{\text { Dec. } 1,2006}\) start, 11 percent of teen drivers were observed using cell phones as they left school in the afternoon. Abour 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) can be regarded as representative of these populations of teen drivers. Suppose also that 200 teen drivers were observed before the ban (so \(n_{1}=200\) and \(\hat{p}_{1}=.11\) ) and 150 teen drivers were observed after the ban. a. Construct and interpret a \(95 \%\) 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 zero included in the confidence interval of Part (c)? What does this imply about the difference in the population proportions?

Step by step solution

01

Calculate the proportions

Firstly, the observed proportions of teen drivers using cellphones before and after the ban need to be calculated. Given that \(n_{1}=200\) and \(\hat{p}_{1}=.11\) denoting the sample size and proportion before the ban, we know \(n_{2}=150\) teens were observed after the ban but do not have \(\hat{p}_{2}\), the proportion for this group, which can be found by dividing the number of teen drivers observed using cellphones after the ban by the total number observed after the ban.

02

Construct the confidence interval

A 95% confidence interval for the difference in the proportions (before and after the ban) can be calculated using this formula:\[(\hat{p}_{1} - \hat{p}_{2}) \pm Z \sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1})}{n_{1}}+\frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_{2}}}\]Where:\(\hat{p}_{1}\) and \(\hat{p}_{2}\) are the observed proportions of teens using cellphones before and after the ban, respectively,\(n_{1}\) and \(n_{2}\) are the sizes of the observational groups before and after the ban, respectively, and \(Z\) is the z-value from the standard normal distribution corresponding to the desired level of confidence (for a 95% confidence level, \(Z = 1.96\)).

03

Interpret the confidence interval

The calculated confidence interval denotes the range in which we are 95% confident that the actual difference in proportions (population parameters) lies.

04

Inclusion of zero

Determining whether zero is included in the confidence interval allows us to infer whether there was a significant difference in the proportions. If the interval includes zero, it indicates that there is no significant difference between the proportions. If it does not include zero, it suggests that there is a significant difference between the proportions.

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Key Concepts

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

Proportions

In statistics, a proportion is a statistical measure that indicates the fraction or percentage of a sample or population that possesses a certain characteristic. In our case, we're looking at the proportion of teen drivers using cell phones before and after a specific ban was implemented.
This is represented by \( \hat{p} \), where a subscript usually distinguishes between different groups, such as \( \hat{p}_{1} \) for the proportion before the ban, and \( \hat{p}_{2} \) for after the ban.

The calculation of these sample proportions is crucial. It's done by dividing the number of individuals in the sample with the characteristic of interest by the total number in the sample. For instance, if 22 out of 200 observed teens were using phones before the ban, then \( \hat{p}_{1} = \frac{22}{200} = 0.11 \). In this exercise, the proportion after the ban needs to be calculated similarly. Proportions form the foundation for understanding changes or differences between groups.

Population Parameters

A population parameter is a value that represents some characteristic of the entire population. While a sample provides estimates that help us infer these parameters, the true value usually remains unknown.
In this exercise, we are focusing on two population parameters - the proportion of teen drivers using cell phones before and after a ban was implemented.

These proportions are population parameters that we estimate using our sample data. Confidence intervals are constructed around these sample proportions to give us a range of possible values for the population parameter. The goal is to see how likely it is that the true parameter lies within this interval.

• This means we can say with a certain level of confidence (e.g., 95%) that the true difference in proportions before and after the ban falls within our calculated interval.
• It's important to remember that we're using sample data to estimate these parameters, which involves a level of uncertainty.

An understanding of population parameters enables effective decision-making based on statistical estimation.

Significant Difference

When conducting statistical analysis, determining whether a difference is significant is foundational. In the context of this exercise, we are computing whether the difference in proportions of teen drivers using phones is statistically significant after the ban was in place.

The confidence interval for the difference between two proportions is key here. Because this interval indicates a range of values for the true difference, examining whether 'zero' falls into this interval is crucial in determining significance.

• If zero falls within the interval, it suggests there is not enough evidence to confirm that the ban had an effect, i.e., there's no significant difference.
• Conversely, if zero is not within the interval, it suggests a significant difference does exist. This implies the ban had a tangible effect on phone use among teen drivers.

Significant differences help policy makers, like those involved with the cell phone ban, understand the effectiveness of their interventions, ultimately aiding in crafting strategies to improve public safety.