91Ó°ÊÓ

A "social experiment" conducted by a TV program questioned what people do when they see a very obviously bruised woman getting picked on by her boyfriend. On two different occasions at the same restaurant, the same couple was depicted. In one scenario the woman was dressed "provocatively" and in the other scenario the woman was dressed "conservatively". The table below shows how many restaurant diners were present under each scenario, and whether or not they intervened. \begin{tabular}{lccc} & \multicolumn{2}{c} { Scenario } & \\ \cline { 2 - 3 } & Provocative & Conservative & Total \\ \hline Yes & 5 & 15 & 20 \\ No & 15 & 10 & 25 \\ \hline Total & 20 & 25 & 45 \end{tabular} A simulation was conducted to test if people react differently under the two scenarios. 10,000 simulated differences were generated to construct the null distribution shown. The value \(\hat{p}_{p r, s i m}\) represents the proportion of diners who intervened in the simulation for the provocatively dressed woman, and \(\hat{p}_{c o n, s i m}\) is the proportion for the conservatively dressed woman. (a) What are the hypotheses? For the purposes of this exercise, you may assume that each observed person at the restaurant behaved independently, though we would want to evaluate this assumption more rigorously if we were reporting these results. (b) Calculate the observed difference between the rates of intervention under the provocative and conservative scenarios: \(\hat{p}_{p r}-\hat{p}_{c o n}\) (c) Estimate the p-value using the figure above and determine the conclusion of the hypothesis test.

Step by step solution

01

Define the Hypotheses

We need to establish the null and alternative hypotheses for the scenario. The null hypothesis (\(H_0\)) states that there is no difference in the proportion of diners who intervene between the two clothing scenarios: \(H_0: \hat{p}_{pr} = \hat{p}_{con}\). The alternative hypothesis (\(H_a\)) posits that the proportions are different: \(H_a: \hat{p}_{pr} eq \hat{p}_{con}\).

02

Calculate the Observed Proportions

Calculate the proportion of diners who intervened in both scenarios. For the provocative scenario, \(\hat{p}_{pr} = \frac{5}{20} = 0.25\). For the conservative scenario, \(\hat{p}_{con} = \frac{15}{25} = 0.60\).

03

Determine the Observed Difference

Compute the observed difference in proportions: \(\hat{p}_{pr} - \hat{p}_{con} = 0.25 - 0.60 = -0.35\).

04

Estimate the P-value

A p-value is estimated based on the null distribution curve from the simulation data. We compare the observed difference of -0.35 against these simulations to find how often such a difference occurs. If very few simulations result in a difference as extreme as -0.35, the p-value will be low.

05

Draw a Conclusion from the P-value

If the p-value is less than the significance level (usually 0.05), we reject the null hypothesis, suggesting a significant difference in intervention rates. If the p-value is above 0.05, we fail to reject the null hypothesis, indicating insufficient evidence to claim a significant difference.

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Key Concepts

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

Null Hypothesis

A null hypothesis, denoted as \( H_0 \), is a starting assumption in statistical testing. For this exercise, the null hypothesis is that there is no difference in the proportions of intervention between the two scenarios presented in the experiment. This means that whether the woman is dressed provocatively or conservatively, the rate at which diners intervene remains consistent. We express this hypothesis mathematically as \( H_0: \hat{p}_{pr} = \hat{p}_{con} \).
The null hypothesis serves as a baseline or "no effect" benchmark against which observations are measured. If the data significantly deviates from what the null hypothesis predicts, this suggests that the null hypothesis might not hold true.

Alternative Hypothesis

The alternative hypothesis, denoted as \( H_a \), contrasts the null hypothesis by proposing that there is a difference between the rates of intervention in the two scenarios. In this exercise, the alternative hypothesis is that the proportions of intervention are not equal: \( H_a: \hat{p}_{pr} eq \hat{p}_{con} \).
This hypothesis provides the basis for further investigation and is accepted if the null hypothesis is rejected. It's essential in hypothesis testing because it drives the purpose of the study, which in this case is to determine whether clothing affects intervention behavior.

• This hypothesis implies some form of association or effect.
• It is what researchers often seek to support through their tests.

P-Value Estimation

The p-value is a crucial statistical concept used to interpret the results of hypothesis testing. It represents the probability of observing a result as extreme as the one obtained, assuming the null hypothesis is true. In this context, our calculated observed difference in proportions is \(-0.35\).
Estimating the p-value involves comparing the observed difference to a simulated null distribution. If the null hypothesis is valid, few simulated differences should be as large or larger in magnitude than observed.

• A low p-value (typically less than 0.05) indicates that such an extreme observed difference is unlikely, thus we reject the null hypothesis.
• A high p-value suggests that the observed difference is consistent with the null hypothesis.

Independent Observations

The assumption of independent observations is foundational in statistical testing to ensure valid, unbiased results. For this exercise, we assume that each diner's response at the restaurant is independent of the others. This means that the decision of one person to intervene or not doesn't influence another's decision.
Why is this assumption important?

• It safeguards the integrity and validity of the statistical results.
• Dependency among observations could distort the analysis and lead to incorrect conclusions.

In real-world scenarios like this, testing independence is not straightforward, but researchers should consider potential dependencies between observations and account for them in their analysis if suspected.