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The paper "Passenger and Cell Phone Conversations in Simulated Driving" (Journal of Experimental Psychology: Applied [2008]: \(392-400\) ) describes an experiment that investigated if talking on a cell phone while driving is more distracting than talking with a passenger. Drivers were randomly assigned to one of two groups. The 40 drivers in the cell phone group talked on a cell phone while driving in a simulator. The 40 drivers in the passenger group talked with a passenger in the car while driving in the simulator. The drivers were instructed to exit the highway when they came to a rest stop. Of the drivers talking to a passenger, 21 noticed the rest stop and exited. For the drivers talking on a cell phone, 11 noticed the rest stop and exited. a. Use the given information to construct and interpret a \(95 \%\) confidence interval for the difference in the proportions of drivers who would exit at the rest stop. b. Does the interval from Part (a) support the conclusion that drivers using a cell phone are more likely to miss the exit than drivers talking with a passenger? Explain how you used the confidence interval to answer this question.

Step by step solution

01

Calculate the proportions

First, we need to calculate the proportions of drivers who noticed the rest stop in each group. The proportion for the passenger group is \(p_{1} = \frac{21}{40}=0.525\). The proportion for the cell phone group is \(p_{2}=\frac{11}{40}=0.275\). The difference between the two proportions is \(p_{1}-p_{2}=0.525-0.275=0.25\).

02

Calculate the standard error

The standard error for the difference between two proportions can be calculated using the formula: \[\sqrt{\frac{p_{1}(1-p_{1})}{n_{1}} + \frac{p_{2}(1-p_{2})}{n_{2}}}\] Substituting the values we have: standard error = \[\sqrt{\frac{0.525(1-0.525)}{40} + \frac{0.275(1-0.275)}{40}}\] After the calculation, the standard error is approximately 0.102.

03

Calculate the confidence interval

A \(95\%\) confidence interval for the difference between two proportions is calculated as \[p_{1}-p_{2} \pm Z_{\alpha/2}\times\text{standard error}\] The value for \(Z_{\alpha/2}\) for a \(95\%\) confidence interval is approximately 1.96. Plugging this into the formula along with the other values, we get: \[0.25 \pm 1.96(0.102)\] This provides a confidence interval of approximately \((0.05, 0.45)\).

04

Interpret the results

The calculated \(95\%\) confidence interval is from \(5\%\) to \(45\%\), indicating that we are \(95\%\) confident that the true difference in proportions (proportion of drivers speaking with a passenger - proportion of drivers speaking on a phone) falls in this interval. That is, drivers talking to passengers are between \(5\%\) and \(45\%\) more likely to notice the rest stop compared to those talking on cell phones.

05

Answer the second question

The interval entirely lies above zero, it supports the conclusion that drivers using a cell phone are more likely to miss the exit than drivers talking with a passenger. The positive difference means that the proportion of passenger group who noticed the stop is higher than on the phone group, therefore those on phones are more likely to miss the stop.

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Key Concepts

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

Proportion Comparisons

When comparing two groups in a statistical study, proportions serve as a powerful tool to understand the relative frequencies of an event occurring in each group. In the context of the simulated driving experiments, where one group talked on a cell phone and the other with a passenger while driving, proportion comparisons allow us to measure the impact of these two different conditions on driver behavior. Specifically, we calculate the proportion of drivers who noticed the rest stop in each scenario, which is simply the number of drivers who exited the simulator divided by the total number of drivers in that group. By comparing these proportions, we can begin to deduce the relative level of distraction caused by each type of conversation.

Standard Error Calculation

To understand the precision of our proportion comparisons, we calculate the standard error of the difference between two proportions. This is crucial as it reflects the amount of sampling variability and gives us an idea of how much the proportion difference we observed might fluctuate if we were to repeat the experiment. The standard error in this context is calculated using a specific formula that incorporates the proportions of each group along with their respective sample sizes. It quantifies the potential variance between the observed difference and the real, but typically unknown, difference in the population.

Interpretation of Confidence Intervals

Confidence intervals are a cornerstone of inferential statistics, used to estimate the range in which a population parameter is believed to reside, based on sample data. For our driving simulation study, we interpreted a 95% confidence interval for the difference in the proportions of drivers who noticed the rest stop. This interval tells us that, if we were to repeat the experiment many times, 95% of the calculated intervals would contain the true difference in proportions. It is a measure of the reliability of an estimate. In the driving study, the fact that the confidence interval does not include zero strengthens our belief in a real difference between the two groups.

Simulated Driving Experiments

Simulated driving experiments are a powerful method for studying behavior in a controlled environment that mimics real-world driving conditions. This type of experiment helps us gain insights into driver performance and response under various scenarios, such as conversing with a passenger or on a cell phone. The advantage of simulations is that they can be conducted without the risk associated with real-life driving experiments, while still providing valuable data that can be analyzed using statistical methods to draw meaningful conclusions about driver behavior.

Statistical Significance

Determining statistical significance is fundamental in assessing whether the observed difference between groups is due to a genuine effect or merely random chance. In the driving simulator study, we look at whether the proportion of drivers who missed the exit while talking on a cell phone is significantly higher than those conversing with a passenger. Statistical significance is often tested using a pre-determined significance level (commonly 0.05) and comparing it with a p-value or examining confidence intervals. If the confidence interval for the difference in proportions does not contain zero (as in our study), we infer that there is significant evidence to suggest a real difference exists, implying that the observed effect is unlikely to be due to chance.