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Stopping Traffic with a Smile! Throughout Europe, more than 8000 pedestrians are killed each year in road accidents, with approximately \(25 \%\) of them dying when using a pedestrian crossing. Alt hough failure to stop for pedestrians at a pedestrian crossing is a serious traffic offense in France, more than half of drivers do not stop when a pedestrian is waiting at a crosswalk. In this experiment, a male research assistant was instructed to stand at a pedestrian crosswalk and stare at the driver's face as a car approached the crosswalk. In 400 trials, the research assistant maintained a neutral expression, and in a second set of 400 trials, the research assistant was instructed to smile. The order of smiling or not smiling was randomized, and several pedestrian crossings were used in a town on the coast in the west of France. The research assistants was dressed in normal attire for his age (jeans, t-shirt, and sneakers). In the 400 trials in which the assistant maintained a neutral expression, the driver stopped in 172 out of the 400 trials, while in the 400 trials in which the assistant smiled at the driver, the driver stopped 226 times. 2 Do a test to assess the evidence that a smile increases the proportion of drivers who stopped. (In the portion of the study using female assistants, a smile significantly increased the proportion of drivers who stopped, with the proportion who stopped being significantly higher than for males in both the neutral and smiling conditions.)

Step by step solution

01

Define Hypotheses

To assess if a smile increases the proportion of drivers who stop, define the null and alternative hypotheses. - Null Hypothesis, \(H_0\): There is no difference in the proportions of drivers stopping when the assistant smiles versus when they maintain a neutral expression. Mathematically, \( p_1 = p_2 \), where \( p_1 \) is the proportion of drivers who stop when smiled at, and \( p_2 \) is the proportion when not. - Alternative Hypothesis, \(H_a\): The proportion of drivers who stop increases when the assistant smiles. Mathematically, \( p_1 > p_2 \).

02

Calculate Sample Proportions

Calculate the sample proportions of drivers stopping in each scenario. \[\hat{p_1} = \frac{226}{400} = 0.565\] \[\hat{p_2} = \frac{172}{400} = 0.43\] Where \(\hat{p_1}\) is the proportion who stopped when smiled at, and \(\hat{p_2}\) is the proportion who stopped when neutral.

03

Determine Pooled Proportion

Calculate the pooled proportion, \(\hat{p}\), which combines the data from both scenarios:\[\hat{p} = \frac{226 + 172}{400 + 400} = \frac{398}{800} = 0.4975\]

04

Calculate Standard Error

The standard error (SE) of the difference in proportions is calculated as follows:\[SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}\]\[= \sqrt{0.4975 \times (1-0.4975) \times \left(\frac{1}{400} + \frac{1}{400}\right)}\]\[= 0.0354\]

05

Calculate Z-Score

To find the Z-score, calculate the difference in sample proportions divided by the standard error:\[Z = \frac{\hat{p_1} - \hat{p_2}}{SE}\]\[Z = \frac{0.565 - 0.43}{0.0354} \approx 3.82\]

06

Determine Statistical Significance

Using the standard normal distribution, compare the Z-score to a critical value (typically 1.645 for a one-tailed test at 0.05 significance level). Since \(Z = 3.82\), which is greater than 1.645, we reject \(H_0\). There is significant evidence to suggest that drivers are more likely to stop when the assistant smiles.

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Key Concepts

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

Proportions

Proportions are a way to express a part of a whole. In hypothesis testing, they help us understand how often a certain outcome occurs compared to the total number of possibilities. In the context of the experiment about driver behavior, the proportion describes how many drivers stopped at a crosswalk when a pedestrian either smiled or maintained a neutral expression. The formula for calculating a proportion is simple: \( \hat{p} = \frac{x}{n} \), where \( x \) is the number of successes (drivers who stopped) and \( n \) is the total number of trials.

• For the group that was smiled at, the proportion \( \hat{p_1} \) is \( \frac{226}{400} = 0.565 \).


• For the neutral group, the proportion \( \hat{p_2} \) is \( \frac{172}{400} = 0.43 \).

These proportions provide a basis for comparing the effectiveness of smiling at drivers versus not smiling.

Null and Alternative Hypotheses

In hypothesis testing, we begin by establishing a null hypothesis and an alternative hypothesis. These hypotheses are essential in determining if the observed data can support a specific claim.

• Null Hypothesis, \(H_0\): In this experiment, the null hypothesis states there is no difference in stopping behavior between the scenarios where the driver is smiled at versus when a neutral expression is maintained. Symbolically, this is \( p_1 = p_2 \).


• Alternative Hypothesis, \(H_a\): The alternative hypothesis asserts that smiling actually increases the likelihood of drivers stopping, meaning \( p_1 > p_2 \).

The goal of testing these hypotheses is to determine if we can reliably reject the null hypothesis in favor of the alternative, based on the evidence from the collected data.

Z-score

The Z-score in hypothesis testing measures how many standard deviations an element is from the mean of the population. It helps determine whether the observed difference between two proportions is statistically significant.To calculate the Z-score:1. Subtract the neutral proportion \( \hat{p_2} \) from the smiling proportion \( \hat{p_1} \).2. Divide the result by the standard error (SE) of the difference between proportions.For this experiment:\[ Z = \frac{\hat{p_1} - \hat{p_2}}{SE} = \frac{0.565 - 0.43}{0.0354} \approx 3.82 \]This Z-score helps us understand the strength and direction of the difference between the two groups.

Statistical Significance

Statistical significance gives us a way to determine if the observed results in a study are likely due to the effect of an intervention rather than random chance. In our experiment, it's about determining if the smile leads to a significantly higher stopping rate among drivers.To establish statistical significance, we compare the Z-score to a critical value from the standard normal distribution. For a typical one-tailed test with a 0.05 significance level, this critical value is 1.645.Since the calculated Z-score \(3.82\) is greater than 1.645, we reject the null hypothesis \(H_0\). This rejection means there is enough evidence to say that smiling increases the probability that drivers will stop at the pedestrian crossing, achieving statistical significance in our test.