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An American Automobile Association (AAA) study imestigated the question of whether aman or a woman was more likely to stop and ask for directions (AAA, Janury 2006 ). Thesituation referred to in the study stated the following: "If you and your spouse are drivingtogether and become lost, would you stop and ask for directions?" A sample representativeof the data used by AAA showed 300 of 811 women said that they would stop and ask fordirections, while 255 of 750 men said they would stop and ask for directions.a. The AAA research hypothesis was that women would be more likely to say thatthey would stop and ask for directions. Formulate the null and alternative hypotheses for this study.b. What is the percentage of women who indicated that they would stop and ask fordirections?c. What is the percentage of men who indicated that they would stop and ask for directions?d. $A t \alpha=.05$ , test the hypothesis. What is the $p$ -value, and what conclusion would youexpect AAA to draw from this study?

a. $H_{0} : \mu_{1} \leq \mu_{2}, H_{a} : \mu_{1}>\mu_{2}$b. 36.99$\%$c. 34.00$\%$d. $P=0.1131$ , There is not sufficient evidence to support the claimthat women would be more likely to say that they would stop and ask fordirections.

Intro Stats / AP Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

Descriptive Statistics

Confidence Intervals

The Chi-Square Distribution

Piedmont College

Oregon State University

University of St. Thomas

Lectures

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02:09

Men have a reputation for …

02:22

Gender and Direction. In t…

03:19

Preliminary data analyses …

so the first thing we're asked to do is come up with hypotheses. Thio test what these researchers believe, and they thought that the proportion of woman that's that say that they ask for directions it would be greater than the proportion of men. So, um, with that in mind, we're going to say that the the alternative hypothesis what the researchers believed is that the proportion of women is greater than the proportion of men, and this means the proportion that ask for directions. And that means the no hypothesis would just be whatever the opposite of this is. So the portion of women is less than or equal to the proportion of men. Or you could write to these hypotheses as differences of proportions. The proportion of women minus the proportion of men be less than equal to zero for the no or the proportion of women minus. The proportion of men is greater than zero for the alternative hypothesis. So this is our answer to part a important be were asked to find the sample proportion of women that asked for directions. So we're given that we have, um, 811 women that were surveyed And of those 811 women 300 said that they would be willing to ask for questions. It would stop and ask for questions. So we get a sample proportion of, um 0.3699 And so this is our answer to Part B. Um, but if you want to put it in percent forum, we could say 36.99%. And now in part, See, we're just asked you the same thing, but with men, and we get that we have a sample size of 750. And of that, 255 men said that they would stop and ask for directions. So we have, Well, you've 0.34 if we want to put that in percent form 34%. Now we have to use this information to come up with a test statistic and then a Z score and then find a P value to compare to our Alfa our significance. Level two figure out if we should reject or accept the null hypothesis. So, um, I'm just gonna I just wrote this on the side here. Just remember that our alphas 0.5 is, the first thing we have to do is come up with a test statistic because we're working with a difference of proportions, we're going to find the pool variants. Another formula for the pool variance is equal to the sample size and the first population times the proportion the sample proportion in the first population. So in this situation, it would be the sample size of the women's population times the proportion of women plus the sample size, the men's population timesthe sample proportion in the men population over the sum of the populations. Um, but we already computed these or we didn't even need to compute these. We were given these values in the problem. We were given the number of men and women that would, um, say that they would respond. And that's how we discovered our P bar. So what we can do is instead of writing ah, doing this math, we can just put down the values that we were given for the number of men and women who said that they would stop and ask for directions. So over here we have 300 women said they would stop and ask, and 255 men would stop and ask. And down here we have a sum of 7 50 plus 8 11 so that we get a a pool variance of 0.35 five. And now, using this pulled variance, we can come up with a Z score. So our Z score is equal to the difference in our proportions, divided by the square root of our pooled variance times one minus our pool variants times one over our first sample size plus one over our second sample size. So for us, this is equal to 0.36 0.3699 minus 0.34 divided by the square root of our pool variance, which we just figured out is 0.355 times one minus 10.355 times one over our first sample size of 8 11 plus one over our second sample size of 7 50 and we get a Z score of 1.23 before, So now we have to look at this to find the direction of our, um, her hypothesis test. And because we are looking at the difference being greater than zero, we see that This is a right tailed Z test. So we draw our normal curve. Z equals zero lives in the middle. We discovered that we have a Z score of 1.2 three four 12341.234 And, um, we're interested in a rate tailed Z test, so we're looking for this value to the right of Z equals 1.234 But that value is kind of hard to find. But what we can do is find this area the probability that Z is less than 1.234 subtracted from one so probability that Z is greater than or equal to 1.234 is equal to no is equal to one, minus the probability that Z is less than 1.234 And using that logic, we gotta answer of 0.1093 Our P value is 0.1093 And now we can compare a P value to our Alpha of 0.5 Um because, uh, are our P value of 0.1093 it's greater than 0.5 Therefore, we failed to reject the no hypothesis. What does that mean? That means that there is not sufficient evidence to support the claim that women would be more likely to say yes to the survey.

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