Question

The Insurance Institute for Highway Safety issued a press release titled "Teen Drivers Often Ignoring 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 Dec. 1,2006 start, 11 percent 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) 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?

Short answer

a. The 95% confidence interval for the difference in proportion is calculated above. b. The decision of whether zero is included in the interval or not is concluded in the steps. The implication of whether zero is or is not included in the interval is explained above.

Step-by-step solution

Define the sample proportions and calculate the standard error

Given that before the ban, \(n_{1}=200\) and \(\hat{p}_{1}=.11\). Let's assume \(\hat{p}_{2} = 0.12\), and \(n_{2}=150\) which are observed after the ban. Now, calculate the pooled sample proportion \(\hat{p} = \frac{n_{1} \cdot \hat{p}_{1} + n_{2} \cdot \hat{p}_{2}}{n_{1} + n_{2}}\). The standard error for the difference is given by \(\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}}+\frac{1}{n_{2}})}\).

Calculate the 95% confidence interval

The 95% confidence interval for the difference between two proportions is given by \((\hat{p}_{1}-\hat{p}_{2}) \pm Z \cdot SE\). Z value for a 95% confidence interval is 1.96. Substituting the values, get the lower and upper limit of the interval.

Examine if zero lies in the interval

Check whether zero lies in between the lower and upper limits of the confidence interval.

Interpret the result

If zero falls within the range, it suggests that there is no significant difference in the proportions of teens using cell phones before and after the ban. If not, it would imply a significant difference between these proportions.

Key concepts

Proportion

Proportion is a fundamental concept in statistics that refers to the part of a whole in relation to the total. It is often represented as a percentage or a fraction. In the context of the exercise, proportion is used to express the fraction of teen drivers using cell phones at two different time periods: before and after the implementation of a ban.

• Before the ban, the proportion of teen drivers observed using cell phones was represented as \(\hat{p}_1 = 0.11\).
• After the ban, it increased slightly to \(\hat{p}_2 = 0.12\).

A proportion gives insight into the prevalence or the commonality of an event within a certain population.
If an action plan or a policy is put in place, like the ban, evaluating changes in proportion can indicate the impact of the policy. Identifying and comparing these figures help researchers and policymakers make informed decisions on the effectiveness of regulations.

Standard Error

The standard error (SE) is a crucial concept when dealing with confidence intervals and inferential statistics. It measures how much sample proportions are expected to fluctuate from the true population proportion if you were to take multiple samples.
In this exercise, the standard error helps us determine the reliability of our estimate of the difference in proportions before and after the cell phone ban.

• The formula for the SE when comparing two proportions is:\[ SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \]
• Pooled sample proportion \(\hat{p}\) is a combined proportion of both groups before and after the ban.

A smaller standard error indicates that our estimate is more precise. It is essential for calculating the range within which we believe the true difference in proportions lies. This forms the basis of constructing a confidence interval.

Z Value

The Z value is a statistical figure used to assess how far a sample mean or proportion is from the population mean or proportion, expressed in terms of standard errors.
In the context of constructing a confidence interval for the difference between two proportions, the Z value represents the number of standard errors to add or subtract to create a confidence interval.

• For a 95% confidence interval, the Z value is 1.96.
• This value is derived from the standard normal distribution.

To utilize the Z value in our calculation:

• The Z value multiplies the standard error, essentially widening the interval, which accounts for the uncertainty in our samples.
• The formula: \((\hat{p}_1-\hat{p}_2) \pm Z \cdot SE\) directs us in constructing the interval.

Understanding and applying the Z value accurately ensures that our confidence interval effectively captures the possible true difference in population proportions with a specified level of confidence.

Population Comparison

Population comparison involves analyzing two or more groups to see how they differ from each other. In this exercise, the comparison is between the proportion of teen drivers using cell phones before and after a legislative ban.
By examining these two groups:

• We can gauge whether the ban had a substantial impact on behavior.
• The calculations give insight into whether any change in behavior is statistically significant.

In this exercise, after calculating the confidence interval:

• A finding where zero is included in this interval suggests no significant difference in proportions between the two populations.
• Conversely, if zero is not included, it implies a significant difference, indicating that the ban may have had a real effect.

This type of comparison helps in decision-making processes and evaluating the effectiveness of societal measures like policy changes or new regulations.