A "social experiment" conducted by a TV program questioned how people would act 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". We would like to know if the rate of intervention was different based on how the woman was dressed at a $5\%$ level of significance. The table below shows how many restaurant diners were present under each scenario, and whether or not they intervened.

$\backslash$table$[[,$Provocative,Conservative,Total$],[$Yes$,5,15,20],[$No$,15,10,25],[$Total$,20,25,45]]$

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$(ppr),$ represents the proportion of diners who intervened in the simulation for the provocatively dressed woman, and widehat$(p)$ con, $$ is the proportion for the conservatively dressed woman.

a$)$ What are the hypotheses? For the purposes of this e: rcise, 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.

$H_0$:$p_{pr}-p_{\text{c}\text{o}\text{n}},0$

$H_A$:$p_{pr}-$pcon,$0$

b$)$ Calculate the observed difference between the rates of intervention under the provocative and conservative scenarios: widehat$(p)pr-$widehat$(p)con=$

c$)$ Estimate the $p-$value using the figure above.

d$)$ Determine the conclusion of the hypothesis test.

We should not reject the nult hypothesis. The data does not show any significant difference in the way that people react under the two scenarios.

We should reject the null hypothesis. The data provide strong evidence that people react differently under the two scenarios.