91Ó°ÊÓ

Statistics student Hector Porath wanted to find out whether gender and the use of turn signals when driving were independent. He made notes when driving in his truck for several weeks. He noted the gender of each person that he observed and whether he or she used the turn signal when turning or changing lanes. (In his state, the law says that you must use your turn signal when changing lanes, as well as when turning.) The data he collected are shown in the table. $$\begin{array}{|l|l|l|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn signal } & 585 & 452 \\ \hline \text { No signal } & 351 & 155 \\\\\hline & 936 & 607 \\ \hline\end{array}$$ a. What percentage of men used turn signals, and what percentage of women used them? b. Assuming the conditions are met (although admittedly this was not a random selection), find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0, and explain whether this provides evidence that the proportions of men and women who use turn signals differ in the population. c. Another student collected similar data with a smaller sample size: $$\begin{array}{|l|c|c|}\hline & \text { Men } & \text { Women } \\\\\hline \text { Turn Signal } & 59 & 45 \\ \hline \text { No Signal } & 35 & 16 \\\\\hline & 94 & 61 \\ \hline\end{array}$$ First find the percentage of men and the percentage of women who used turn signals, and then, assuming the conditions are met, find a \(95 \%\) confidence interval for the difference in percentages. State whether the interval captures 0 , and explain whether this provides evidence that the percentage of men who use turn signals differs from the percentage of women who do so. d. Are the conclusions in parts \(\mathrm{b}\) and \(\mathrm{c}\) different? Explain.

Step by step solution

01

Calculate the Percentages

For the first table. The percentage of men using turn signals is \( \frac{585}{936} \times 100 = 62.50\% \) and for women, it is \( \frac{452}{607} \times 100 = 74.47\% \)

02

Compute Confidence Interval

Calculate a 95% confidence interval for the difference in proportions. Firstly, let's find the pooled proportion, \( p = \frac{585 + 452}{936+607} \). The standard error can be computed using the formula: \( SE = \sqrt{p(1-p)(\frac{1}{936} + \frac{1}{607})} \), and, 95% confidence interval would be \( p_{1} - p_{2} \pm Z_{\alpha/2} \times SE \) where \( p_{1} \) and \( p_{2} \) are the proportions of men and women respectively and \( Z_{\alpha/2} \) is the critical value from the standard normal distribution table for a 95% confidence interval, which is 1.96.

03

Analyze Confidence Interval

Check if the confidence interval captures 0 to see if there's a difference between the proportions of men and women who use turn signals. If the confidence interval contains 0, then there is no significant difference between the proportions of men and women who use turn signals. If the interval does not contain 0, then there is a significant difference.

04

Repeat Steps for New Data

Repeat steps 1-3 for the new set of data given in part c using the formulas mentioned in the steps.

05

Compare Findings

Compare the conclusions from part b and c by analyzing their respective 95% confidence intervals. Determine if the percentages and confidence intervals are similar or not to conclude if the findings are different.

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Key Concepts

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

Confidence Interval

When performing statistical analysis, one fundamental concept is the confidence interval (CI). This is a range of values, derived from data, which is believed to contain the true value of an unknown population parameter with a certain level of confidence, typically expressed as a percentage such as 95% or 99%. In the context of Hector Porath's traffic study, a confidence interval for the difference in proportions of turn signal usage between men and women provides information about whether the observed difference is statistically meaningful, considering random variation that could occur in samples.

To construct a CI for the difference between two proportions, as in Hector's investigation, we first require the sample proportions. For example, the solution indicates that the percentage of men using turn signals is approximately 62.50%, while for women it is 74.47%. Using the pooled proportion method, where we combine both samples to estimate a common proportion, we calculate the standard error of the difference. The 95% CI is then constructed by taking the difference in the sample proportions and adding and subtracting the product of the standard error and the Z-score, which corresponds to the desired confidence level. This interval offers valuable insight; if it contains zero, it suggests that the true population difference might be non-existent (no signal usage difference between genders), whereas an interval that doesn't include zero implies a meaningful difference does exist.

Statistical Significance

A closely related concept to the confidence interval is statistical significance. This term indicates whether the difference observed in a study is likely due to chance or if it reflects a true effect in the population. In statistical hypothesis testing, an effect is considered statistically significant if the p-value is less than a pre-specified significance level, often set at 0.05 (or 5%).

In the scenario presented, Hector is evaluating the statistical significance of the difference in turn signal usage by gender. By calculating the 95% confidence interval for this difference and observing whether it includes zero, Hector can infer statistical significance. If the CI does not include zero, it suggests that the difference in turn signal usage between men and women is significant; the likelihood that such a substantial difference could be observed by random chance alone is low, typically below the 5% threshold.

Data Analysis

Finally, data analysis encompasses processing and interpreting data to extract meaningful information. In Hector's study, data analysis involves several steps: collecting the data, calculating percentages, determining confidence intervals, and evaluating statistical significance. Effective data analysis goes beyond just crunching numbers; it allows us to make informed decisions based on empirical evidence.

Improving the data analysis process might involve examining the assumptions underlying the statistical tests, such as sample size, randomness, and sampling distribution. Considering these factors is crucial for ensuring accurate and reliable interpretations. For example, while Hector's findings from a larger dataset might provide a stable estimate of the true effect, a smaller dataset, like the one collected by another student, might yield less precise estimates and wider confidence intervals, deserving cautious interpretation. Comparing the findings from larger and smaller samples can shed light on the variability of the results and the reliability of the conclusions drawn from the data.