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Throughout Europe, more than 8000 pedestrians are killed each year in road accidents with approximately \(25 \%\) of these dying when using a pedestrian crossing. Although 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. Research assistants were dressed in normal attire for their age (jeans/T-shirt/trainers/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. \({ }^{26}\) Do a test to assess the evidence that a smile increases the proportion of drivers who stop. (In the portion of the study reported in Exercise \(22.30\) 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 the Hypotheses

To assess if a smile influences the proportion of drivers stopping, start by setting up the null and alternative hypotheses. - Null Hypothesis (\(H_0\)): There is no difference in the proportion of drivers who stop when the research assistant smiles.- Alternative Hypothesis (\(H_a\)): There is an increase in the proportion of drivers who stop when the research assistant smiles.

02

Calculate the Proportions

Determine the proportions of drivers stopping in both scenarios:- Neutral Expression: \(\hat{p}_1 = \frac{172}{400} = 0.43\)- Smiling: \(\hat{p}_2 = \frac{226}{400} = 0.565\)

03

Calculate the Difference in Proportions

Compute the observed difference in stopping proportions:\[\hat{p}_2 - \hat{p}_1 = 0.565 - 0.43 = 0.135\]

04

Construct the Pooled Proportion

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

05

Calculate the Standard Error

Calculate the standard error for the difference in proportions using:\[SE = \sqrt{\hat{p}(1-\hat{p})\left(\frac{1}{n_1} + \frac{1}{n_2}\right)}\]\[SE = \sqrt{0.4975(1-0.4975)\left(\frac{1}{400} + \frac{1}{400}\right)}\]\[SE = \sqrt{0.4975 \times 0.5025 \times \frac{2}{400}}\]\[SE \approx 0.03536\]

06

Compute the Z-score

Calculate the Z-score for the difference in proportions:\[Z = \frac{\hat{p}_2 - \hat{p}_1}{SE} = \frac{0.135}{0.03536} \approx 3.82\]

07

Determine the P-value

Use a standard normal distribution table to find the p-value associated with the computed Z-score. A Z-score of 3.82 corresponds to a very low p-value (less than 0.001), which indicates strong evidence against the null hypothesis.

08

Conclusion

Compare the p-value to the significance level (typically \(\alpha = 0.05\)). Since the p-value is much lower than 0.05, we reject the null hypothesis and conclude that a smile significantly increases the proportion of drivers who stop at pedestrian crossings.

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Key Concepts

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

Proportions

Proportions are an essential concept in statistics used to describe a part of a whole. Here, we talk about the proportion of drivers who stop at a pedestrian crossing, either when the research assistant smiles or maintains a neutral expression.
The proportions give us a snapshot of behavior in each scenario. Proportions can be calculated by dividing the number of successful outcomes by the total number of trials.

• For a neutral expression, the proportion is calculated as \( \hat{p}_1 = \frac{172}{400} = 0.43 \).
• For a smiling expression, it is \( \hat{p}_2 = \frac{226}{400} = 0.565 \).

These figures represent how often drivers stopped in response to each behavior. Understanding these proportions is crucial because they form the basis for further statistical analysis, such as comparing multiple groups to assess the effect of different conditions.

Null Hypothesis

When conducting hypothesis testing, the null hypothesis (\( H_0 \)) is a statement that assumes no effect or difference between the groups being compared. It's an essential step in the decision-making process in statistics.
In the context of this study, the null hypothesis states that there is no difference in the proportion of drivers who stop when the research assistant smiles compared to when they exhibit a neutral expression. In simpler terms, \(H_0\) suggests that smiling has no effect on driver behavior at pedestrian crossings.
The null hypothesis is usually tested against an alternative hypothesis (\( H_a \)), which proposes a significant effect or difference. In this case, \( H_a \) suggests that smiling increases the proportion of drivers stopping. The results of the analysis will either lead us to reject or fail to reject the null hypothesis, depending on the evidence provided by the data.

Z-score

The Z-score is a measure that tells us how many standard deviations an element is from the mean. In hypotheses testing for proportions, it helps determine if the observed difference between two groups is statistically significant.
To find the Z-score, we take the difference between the two sample proportions and divide it by the standard error of the difference. Here, we calculate:

• Observed difference: \( \hat{p}_2 - \hat{p}_1 = 0.135 \)
• Standard error: \( SE \approx 0.03536 \)

Thus, the Z-score is calculated as:\[ Z = \frac{0.135}{0.03536} \approx 3.82 \]This high Z-score indicates that the difference in driver stoppage observations with a smile versus a neutral expression is considerably above what could be expected by random chance alone.

P-value

The P-value is a crucial statistic that helps us understand the strength of our results. It tells us the probability of obtaining results as extreme as, or more extreme than, those observed, assuming the null hypothesis is true.
A low P-value indicates that it's unlikely for such results to occur due to random variation alone, providing strong evidence against the null hypothesis.
In this analysis, the Z-score of 3.82 corresponds to a very low P-value (less than 0.001). This means the probability that the observed difference (or a more extreme one) happens under the null hypothesis is less than 0.1%.

• Low P-value (<0.05) allows us to reject the null hypothesis.
• Here, it supports the conclusion that smiling significantly increases the proportion of drivers who stop.

Thus, the P-value is a crucial indicator of the significance of our test results, ensuring informed decisions based on data rather than chance.