Problem

A large automobile insurance company selected sam…

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Problem 29 Hard Difficulty

Jupiter Media used a survey to determine how people use their free time. Watching television was the most popular activity selected by both men and women (The Wall Street Journal, January $26,2004 )$ . The proportion of men and the proportion of women who selected watching television as their most popular leisure time activity can be estimated from the following sample data.
$\begin{array}{ll}{\text { Gender }} & {\text { Sample size }} & {\text { Watching Television }} \\ {\text { Men }} & {800} & {248} \\ {\text { Women }} & {600} & {156}\end{array}$
a. State the hypoctheses that can be used to test for a difference between the proportion
for the population of men and the proportion for the population of women who selected watching television as their most popular leisure time activity.
b. What is the sample proportion of men who selected watching television as their most popular leisure time activity? What is the sample proportion of women?
c. Conduct the hypothesis test and compute the $p$ -value. At a .05 level of significance,
d. What is the margin of error and 95$\%$ confidence interval estimate of the difference between the population proportions?

Answer

a. $H_{0} : p_{1}=p_{2}, H_{a} : p_{1} \neq p_{2}$
b. $0.31,0.26$
c. There is sufficient evidence to support the claim of a difference between
the population proportions.
d. $0.0475,0.0025$ to 0.0975

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Video Transcript

right. So today we want to solve this problem where we want to check if the proportion of men and the proportion of women who selected watching television as they're more spoke color, leisure time activity differs. So in part, a have to kind of set up H not, And each one the no versus alternative hypothesis. And under the new, we will assume that there is no difference in the proportions, Uh, respects to males versus females. Okay, so I'm going to say the proportion of males, it's just the same as the proportion of females right under the alternative, which is a research question. Sometimes they noted US age suit A. We actually want to see their different if the proportions are different. All right, so this is very important because this is going to kind of lead, uh, the way that we're going to compute the p value later cape. So in part B, they're asking us to compute the sample proportion of males on the sample proportion of females who selected a watching television as their most popular leisure time activity. So the book, the sample proportion is the noted US P bar. So in this case, p bar sub m Will they know that the sample proportion for males And this is the number of males who actually Celik watching television, which is 248 uh, divided by the sample size. So we have 800 right? And that these will give us a 0.31 so we can do the same thing for the sample. Proportion of females denoted us P bar some f. Right? So here we have 156 females, um, selected, ah, watching television based on a sample of 600. So these give us, uh, 0.26. So we're done with A and B. Now we can move to see on C. They want us to compete the P value for these hypothesis test. So as we go notice that the alternative hypothesis is still any that these two proportions are not the same. Okay, so that means that it could be greater or less. That means that we're working with both tails, so we have to take into counties in our calculations for the people. Okay, so in C, we went to get, uh, the P value. So a graphical representation of these problem will be, um, that under h note. I am saying that there's no difference. They are the same. Right on. Based on my sample, I can actually find out. What is the sample proportion of males? My nose, the sample proportional females. And we can see that's 0.31 minus 0.26. That's point zero five. And anyway, so I want to see our base in this. If I have enough evidence to reject age not or do not reject age not so I need to convert to these two Aceval. Okay. I need to get a Z volley over here. Okay, So what is the formula for the easy volley? So the formula for the Z value is year point estimate, which is, um, Europea Bar and minus p bar. Uh, female. Okay. And minus devalue under h not, uh, under age, not. I'm saying that the proportion off males mind is the proportion of females, right? This is a value under age not. I am saying that this is zero right, and I will divided by the standard error. Thank you. So the standard error will be the square root off. So this is very important under H, Not under the new hypotheses we're saying, right, We assuming that, uh, these two proportions, the males and females are the same. They're equal to say it to a P volley. Okay, so they're the same. So under each not. I'm saying they're the same. So I have samples, so I need to compute a pooled estimator off these volumes. All right, so this is only the case when under H not. I am assuming that these two proportions are the same. So these pool estimator off P, we will just call it P bar. Okay, Uh, in the standard air will be P bar my nose. Times one minus p. Bar on times one over the sample size off. In this case, males mass plaice, one over the sample size for females. Okay, so this is the formal. So we need to find out this p bar. Everything else is given by the problem. So GNP bar is just the pool of estimation. Estimate estimator off peak. Right. So we're saying underage. Not that these two proportions are this same As a result, we need to find these pull estimator of people. So it's very simple key bar. He's just think I don't have enough space, so let me Can I go? It'll be down. So these people are He's just the sample size of males times uh, sample proportion of males Plas the sound close eyes females times this sample proportion off famous divided by this some off the sample size of males, plus the sample size of pupils. Okay, so, um, bludgeoning some Bali's, we know that we have 800 males. We also know that the sample proportion off males is 0.31. We know that the sample size of females is 600. Assemble proportion of females this 0.26. This is divided by the sum off 800 plus 600 which is 1400 jay. So today, Bali for P bar in this case is zero point 29 07 So this is required to get this Z value This test statistic, right? We're trying to get p value, but before we need to get dizzy. And as we look for the formula for Z, we have these people are everything else is given. So now I am able to cannot compute. Dizzy volley came um So let me Can I go ahead and compute dizzy value? So over here, my Z volley, it's going to be the difference between the sample proportions. With said this 0.31 minus 0.26 minor zero. The difference under age. Now we're saying that there's no difference divided by the standard error. So the standard area is my people are value that we end up having a 0.29 07 times one minus P. Bar zero point 29 07 times, uh, one over the sample size plus one over the sample size. Part one. Over 800. It was one over 600. All of this inside a square root. So eventually we will find out that Z value is, uh, two point zero for right, that's this evil. So going back to the graph. Okay, uh, we find out that dizzy value is 2.0 for, So this is a test of statistics. So from here, where do I go? Were in part C. We're trying to find the p value in order to find the p value. I need to find my test a statistic. My Z body now that I got dizzy value. The question is, where do I go? Check for the alternative hypotheses. It's telling you not equal. So that means that you have to work in the upper tail, not equal implies that it could be greater up until it could be less lower tell. So we have to find these two areas. Can we have to find these two areas? So by driving the standard normal table looking for 2.4 right, we can find thes area on the on the right hand side. Andi and we will see that that area will be a 0.97 93. It's actually the area from here to the left on nineties 0.97 nuclear over here, 97 and 90 cream. Right. That implies that from 2.4 to the right, that is 0.2 07 Okay, But we also have to look for the lower tail. Remember? Is the some of these two So the p value, It's just two times 0.2 07 which is 0.0 for 14. That is the people. So what is the conclusion over here? remember if the P values less than the level of significance in this case, the level of significance is 5% p values less and Alfa level of significance will reject. H not. That is what we can see today. My p value 0.414 He's less than 0.5 Right. So, uh, this is the level of significance. They noted us often, so we rejected. It's not right. We rejected not and conclude that, yes, there is enough evidence that suggests that the proportion off men and the proportion of women who select watching TV as the most popular leisure time activity defense. Actually, the first they're different. So that's all for safe zone D. They're asking us to compute the marching off error. Okay on. They're asking us to estimate a 95% confidence interval for the difference on proportions. Okay, Before we go over there, I notice that here on the easiest statistic on the test statistic the denominator, all these piece from over here is the standard and and I want to use this value in my confidence Interval thesis, standard errors. It's very important because we can actually ah we can actually use it right whenever we compute the margin. Uh, the margin affair. Um Okay, so let me can I go to the next this light and I cannot go for Part two D. And maybe we can make it out here for the standard ever. This is 0.2 for, But Cindy, I want to find out the margin of error on the margin of error. It's define it as a Z value. So Alfa divided by two times this downer air. If I'm looking for a 95% confidence interval, Uh, that implies that AL phase 5% right? This is the area outside a confidence interval divided by two notice. The substrate Alfa, divided by 25% divided by two, is 0.25 times the standard error. So you need to go to the table. Dissertation implies Find me dizzy body such that the operative area is 2.5% but the opportunity is 2.5%. Most of the Z tables all of Then we'll give you, um, areas to the left. Notice that this 0.25 is area on the upper tail. If I want to make a graph, he will imply these shaded area will. The zero point 0 to 5 and I can find is a value where he I'm hoping to find this one. But if I use my table is going to give me the areas to the left. So instead of looking for 25% I have to look inside a table for zero point 975 which is there and to the left and see what are the appropriate columns and rows and you will find out that this is 1.9 for the first common and 0.6 for the first row. If you combine these two, these evil is 1.96. By this standard error was given on part C, right? So now we can find out the margin of error for this problem. And the margin of error is zero point 04 81 Right? So that's my margin affair. Now that I know my margin affair, I can actually compute the call for the 95% confidence interval right on the 95% confidence interval for the difference. The 20 proportions the formula for this case will be the difference between my sample proportions class and minus margin of error. All right, Uh, this difference is just my point. 31 minus 0.26 which is 0.5 on the margin of error was computed as ah 0.0 for 81 So the upper limit is 0.5 plus 0.0 for a one on the lower limit of my confidence. Interval is 0.5 minus 0.4 81. So that gives, says the lower limit on the upper limit. Okay, against again. The lower limit is a 0.5 minus 0.0 for 81 and I end up with zero point 00 19 and the upper limit, 0.5 plus 0.4 81 gives me an upper limit of zero point zero nine eight when we're 95% confidence that the difference between these two proportions are somewhere between 0.190 point 09 81

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