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Before bringing a new product to market, firms carry out extensive studies to learn how consumers react to the product and how best to advertise its advantages. Here are data from a study of a new laundry detergent.29 The participants are a random sample of people who don’t currently use the established brand that the new product will compete with. Give subjects free samples of both detergents. After they have tried both for a while, ask which they prefer. The answers may depend on other facts about how people do laundry.(a) How are laundry practices (water hardness and wash temperature) related to the choice of detergent? Make an appropriate graph to display this relationship. Describe what you see.(b) Determine whether or not the sample provides convincing evidence that laundry practices and product preference are independent in the population of interest.

a. See histogramb. No convincing evidence

Intro Stats / AP Statistics

Chapter 11

Inference for Distributions of Categorical Data

Section 2

Inference for Relationships

Confidence Intervals

The Chi-Square Distribution

Temple University

University of North Carolina at Chapel Hill

University of St. Thomas

Lectures

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05:30

Washing clothes A consumer…

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Use the following informat…

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In June 2007, Consumer Rep…

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Consumer product testing g…

07:06

A study completed by Inter…

13:19

Attitudes toward recycled …

09:14

a. Complete Table 12.31 fo…

All right. So this problem we have we're giving eso data concerning people's water choice or water method of washing their laundry given, um, whether they use the standard or new detergent, it's table up top. I went ahead and already turned that into a table of percentages with respect to their rose. Just cause we're gonna need to do that anyway. And it's just a bunch of calculate calculator work. Yes, we did that. And now we need to graft and summarized these percentages. So using percentages you should have here reverses percents. So let's grab this sense already. Barrier sets All right, And then we need what's our highest percent of highest percent is in the thirties, so I don't need to bother making this higher than five. So 10 20 30 40 50 and there's represents. And then, um, we need to know if they did. Okay, so we'll have standard and new will be so we'll have detergent type, and then we will have I said water. But I guess technically it's your wash type, even if it's the type of water using that defines the wash type. So I guess you could do whatever you want. All right. So standard new 35 31. 35 31. All right. And then we have however seeing type, that was soft water. Um oh, I forget what I labeled them. Crap. Soft water, warm wash. That's what the W was self water. Warm wash for my W's stood for all right And they were gonna have standard new soft water Hot wash 18 and 14 Standard new hot water Warm wash 28 34 and the last one hot water hot wash point to 1.21 All right. And that's Standard, and that is new. So there is our graph, sort of my writings. Little sloppy. It looks like I had more room than I thought I would. I was prepared for this to be really cramped, but it's actually not too bad. Anyway, we're supposed to now summarize our results, so let's look at this graph. If we look at this graph, it looks like, um, well, there's a couple things. It looks like people were most likely to use the standard people. We're most likely two years Standard detergent if they also used soft water. Hot wash awaited the wrong one tonight. That's not true. If they used pot water warm wash this one. I'm like you messing up my bed. I keep he brining down the initials but looking at the wrong picture when I do, they're most likely to use the standard detergent if they used this one, which is soft water warm wash. There we go by bat, keep mixing myself up. Okay? And they were they we're most likely to use new deterred Vince if they also used all right, new detergent, nutrition. It looks like that one. That's that's like it was gonna be mixed up. Newt. Urgent. That's going. This guy that is going to be the hot water. Warm wash, hot water, warm wash. Okay, there's your highs and lows. So we did that just to make sure they did. They correct this 0.34 should be the highest new, and that's definitely true. And this should be the highest standard, and that's definitely true. So I did that correctly. Now, let's see. Is buying preference independent of wash choice. So for part B, we need to do we wanted to a Chi square test. We need to do some things first. So are no hypothesis. Is that on detergent? Preference is independent. Ah, um, washed preference when I say wife preference for talking about the type of water wash their using and then the alternate hypothesis would just be that the detergent preference is not And boat independence of wash preference. So we just copying, pacing that sentence. Okay, now we need to pictures, so that okay, I mean, how many do the chi squared and additions. Nice work conditions. Okay, so the chi square conditions are random. Well, the data came from hey, random sample to tell us that in the on the set of the problem that it was ah, randomized experiment. So we're taking a word on it that it is indeed random. And then there's independence. Well, the total number of people is 354. It looks like 354 is less or equal to 10% of the population of people who do laundry. So that checks out and allows us to basically treat it as with replacement, even though it's not. And then the last thing is large numbers and that's that are expected count. He'd be greater than or equal to five if we look at our oh, so even done unexpected count yet. So let's due to expected count. So are expected. Count we go standard new. I mean, you are for type. So soft water, warm wash, soft water. Uh, hot wash, hard water, warm wash, hard water, hot wash. And then I'm not gonna find out all the calculations. All of these could just calculator work, but just, like, remind you of how we do this under the 1st 1 to read. So we look here. I have a column total of 1 16 aero total of 1 52 and a in equal to 3 54 So I dio 1 16 times already. Forgot what I say. Times 1 52 That's 1 52 divided by 3 54 That equals approximately 49 point eight. And that's what goes here in this first entry. So that's how we do the congregations for these. If you continue in that method, you get 24 47.2, 30.9, 66.2, 32 62.8 and 41.1. Okay, so we did all those now we need to cut a chi squared. So again there's there's eight of these. So Michael animal out that takes forever. So I'll just do like the 1st 2 So what I'm doing chi squared statistic I'm gonna do observed minus expected. So I have 53 63 53 63. All right, so 53 minus 49.8 squared. Divided by 49.8 plus 63 minus 66.2 squared, divided by 66.2. So what I did here is I did the first this here it's did the first column And then you just continue in this method all the way across until we get to the last column, which is gonna have a boy 30 and 42 30 and 42 30 42 for two minutes, 41.1 squared over 41.1 and 30 minus 30.9 squared, divided by 30.9. So it is add up and destroys to remind you so this guy this was the last column here. So we just Adelaide terms. You should have eight fractions here and the end result. It's like high squared is equal to 2.7 How many degrees of freedom are there? Well, degrees of freedom. There are three degrees of freedom and yes, so true freedom equals three. And so if we throw this into you could use the table. I just went ahead and skip straight to using a CDF calculator. I find that the P value when chi squared is 2.7 degrees of freedom of three is 0.55 a one? I don't know. I said value Okay, SP values 0.55801 which is much greater than 0.5 So we thus we fail to reject? No, but says, um and thus we conclude detergent preference to be independent of, uh, wash type preference. And in fact, we significantly rejecting the hypothesis because 0.55 81 is a very large number, which implies there's a lot of evidence against the normal boxes, which, if we look at, uh, the Ark Araff, it kind of makes since that Well, I guess the graph isn't the best. A graph makes it look like it's a lot, but if we look at our expected counts versus observe counts like So my my observed, I'm getting, like, 53 63 in this first column right here. And if I go down to my expected counts, I have 49.8 and 66.2. So, um, those numbers are pretty close, you know, it's pretty small differences there. There's no like giant contributors that would, like throw off the chi squared statistic. And I guess now I could go back to graph. If we look at these tabs in the graph individually notice that they're all almost level. Our idea is that the detergent has nothing to do with the choice. And so the if they were different, what we would have is we would have something like where there is this huge variation across standard and new, where clearly the preference of what they're doing is changing. Um is changing by water type and also by standard or new. That's not the case here. These air almost level, you know, that there's level. Those was well off, like these air almost level. And so since they're almost level, you can see that the water choice seems to be its own thing, and whether or not you're using standard new detergent doesn't seem to make much of a difference on that choice. So it supports our, um, statistic test here.

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