before-bringing-a-new-product-to-market-firms-carry-out-extensive-studies-to-learn-how-consumers-rea

Question

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.

Caleb Huber · Accepted Answer

All right. So this problem we have we&#x27;re giving eso data concerning people&#x27;s water choice or water method of washing their laundry given, um, whether they use the standard or new detergent, it&#x27;s table up top. I went ahead and already turned that into a table of percentages with respect to their rose. Just cause we&#x27;re gonna need to do that anyway. And it&#x27;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&#x27;s grab this sense already. Barrier sets All right, And then we need what&#x27;s our highest percent of highest percent is in the thirties, so I don&#x27;t need to bother making this higher than five. So 10 20 30 40 50 and there&#x27;s represents. And then, um, we need to know if they did. Okay, so we&#x27;ll have standard and new will be so we&#x27;ll have detergent type, and then we will have I said water. But I guess technically it&#x27;s your wash type, even if it&#x27;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&#x27;s what the W was self water. Warm wash for my W&#x27;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&#x27;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&#x27;s actually not too bad. Anyway, we&#x27;re supposed to now summarize our results, so let&#x27;s look at this graph. If we look at this graph, it looks like, um, well, there&#x27;s a couple things. It looks like people were most likely to use the standard people. We&#x27;re most likely two years Standard detergent if they also used soft water. Hot wash awaited the wrong one tonight. That&#x27;s not true. If they used pot water warm wash this one. I&#x27;m like you messing up my bed. I keep he brining down the initials but looking at the wrong picture when I do, they&#x27;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&#x27;re most likely to use new deterred Vince if they also used all right, new detergent, nutrition. It looks like that one. That&#x27;s that&#x27;s like it was gonna be mixed up. Newt. Urgent. That&#x27;s going. This guy that is going to be the hot water. Warm wash, hot water, warm wash. Okay, there&#x27;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&#x27;s definitely true. And this should be the highest standard, and that&#x27;s definitely true. So I did that correctly. Now, let&#x27;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&#x27;re taking a word on it that it is indeed random. And then there&#x27;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&#x27;s not. And then the last thing is large numbers and that&#x27;s that are expected count. He&#x27;d be greater than or equal to five if we look at our oh, so even done unexpected count yet. So let&#x27;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&#x27;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&#x27;s 1 52 divided by 3 54 That equals approximately 49 point eight. And that&#x27;s what goes here in this first entry. So that&#x27;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&#x27;s there&#x27;s eight of these. So Michael animal out that takes forever. So I&#x27;ll just do like the 1st 2 So what I&#x27;m doing chi squared statistic I&#x27;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&#x27;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&#x27;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&#x27;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&#x27;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&#x27;t the best. A graph makes it look like it&#x27;s a lot, but if we look at our expected counts versus observe counts like So my my observed, I&#x27;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&#x27;s pretty small differences there. There&#x27;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&#x27;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&#x27;re doing is changing. Um is changing by water type and also by standard or new. That&#x27;s not the case here. These air almost level, you know, that there&#x27;s level. Those was well off, like these air almost level. And so since they&#x27;re almost level, you can see that the water choice seems to be its own thing, and whether or not you&#x27;re using standard new detergent doesn&#x27;t seem to make much of a difference on that choice. So it supports our, um, statistic test here.

Erin Moser · Answer

puts up stat cats in this video, we&#x27;re gonna be designing an experiment. So a consumer group wants to test the effectiveness of a new organic laundry detergent and make recommendations to customers about how to best use a product. They intentionally get grass stains on 30 white T shirts in order to see how, while the detergent will clean them, they want to try the detergent in cold water and in hot water on both the regular and delicates wash cycles design and appropriate experiment indicating the number of factors, levels and treatments explain the role of random ization in your experiment. Okay, so we already know our sample size is 30 white t shirts that they got stained. And what are we manipulating? What is gonna be our factors? Well, we&#x27;re manipulating the temperature of the water, the water temperature, and we are also manipulating the type of wash. So to figure out our levels, we are gonna want Teoh decide what the levels over factors are. So what water temperatures are we gonna use while we&#x27;re gonna use hot and cold and same thing for type of wash? That said, we&#x27;re gonna use the regular wash and the delicate. So those are factors levels. And then what are we gonna measure? We are going to measure stain removal, so it&#x27;s gonna be a response. Variable. Okay, let&#x27;s go to the next page. So I have a little bit more room so that we can figure out our treatments and a nice, visually organized way. So we are going to have dollar kid and regular wash, And then we are also going to have hot water. Oops. Hot water and cold water. So figure out our treatments. I wanna live extra box re sees. So figure out our treatments, which, if we count, we&#x27;re gonna have four. Um, we&#x27;re just going Teoh combined these different categories, so we&#x27;re gonna have a delicate wash with hot water. We&#x27;re gonna have the delicate wash with cold water. We&#x27;re gonna have a regular wash with hot water and a regular wash with cold water. So all right, so we&#x27;re gonna have four treatments, and then let&#x27;s go again. Let&#x27;s use our space down here to do just another little visual organization just for the experimental design. So we&#x27;re gonna have our 30 T shirts and we&#x27;re gonna have our four treatments which I am gonna abbreviate. So delicate, cold, delicate, hot, regular, cold, regular hot. And then at the end, we&#x27;re gonna compare stain removal. All right? And then, as far as delegating thes 30 shirts into these groups, you&#x27;re gonna wanna label the shirt somehow and put them through a random number generator. And then based on those numbers, that&#x27;s how you&#x27;re gonna delegate into each of our four treatment groups. Alright, guys, that&#x27;s it for this video. I hope you learned a lot else. You next time.

Donald Albin · Answer

the data is shown right here. And ah, first thing we&#x27;re supposed to use is we&#x27;re supposed to treat size as the independent variable and cost as the dependent variable, and we&#x27;re supposed to make a scatter plot. So insert chart. Where is it? There it is. And it already came up was a scatter plot. Very nice. Um, cool. So there&#x27;s four there. All right. Second thing, does it appear that there&#x27;s a relationship? And yes, it does appear that there&#x27;s a relationship. As the size gets bigger, the cost gets higher. Um, it&#x27;s not a linear relationship, but it is a relationship. Calculate the linear relationship, so I just double click on this. Ah, where am I? Here. Trendline. Got linear Soumare squared. Use the equation. There&#x27;s an equation, But look at this. That does not match very well. It doesn&#x27;t go through the points very well. The color fit coat correlation coefficient is the square root of 0.9 six to it&#x27;s 0.98 This is strong correlation, so yes, it is significant. So how much would 40 ounces of 40 outsize cost? Well, it&#x27;s going to be hard for me to predict, because forties somewhere in here. Although you know what? That&#x27;s not that hard to predict. There&#x27;s 40 right there. It&#x27;s gonna be about $5. Chris, you know it&#x27;s gonna be more than 32 let me show you something. If you just put in the equation equals, ah. 0.108 times 40 um, plus four point 57 What&#x27;s $5 in two cents. You can use the equation, or you can read it off the graph. Um, what about 90? Well, now all I have to do is just drag this equation down. $5.52. 54 cents. That looks about right. I noticed that the equation updated itself. It&#x27;s the same equation, but it&#x27;s just looking at 90 instead of 40. Um, I already went over this line. Does not look so good. Um, I think a second degree polynomial would be better or log rhythmic or exponential. Um, are there out Liars note. They&#x27;re all very close to the line. So is this valid for predicting a 300 outsize? Um, somewhat valid. Not as valid as a better curve, but so much valid. So if I put in 300 draggers down $7. 81 cents and that&#x27;s about $7.81. What does the slope represent? Willets 0.108 So it is one cent rounding toe one cent per ounce, so every ounce costs one cent.

Matt Just · Answer

Okay, so we have a function here. Awesome. See? Sequel to a minus C over B minus C. Okay, so we have two products A and be so a is the number of features unique to A and B little B is the number of features unique to to this product beam and C is the number of common features. And FFC is the probability of preference of a over B. Just the probability that a customer will prefer product data product. Okay. And so part A says if we&#x27;re going to assume that a has more features, you need to then be and more than just the number of common features Then where you want to conclude that this function is increasing in concave up. Okay, so which means that we want the 1st and 2nd derivatives to be positive. So let&#x27;s find the first derivative. Now. The variable here is C so we use the quotient rule. Hello? Uh huh. The bottom times the derivative at the top minus the top times the derivative of the bottom. Uh, Hoover P minus C squared. Ok, when you simplify this So this is going to become C minus B minus minus plus plus a minus C So the numerator Caesar cancel will just be a minus B. And that&#x27;s good, because that actually shows. So the denominator is positive. The numerator is positive because A is bigger than B. So this is positive. So the derivative is positive. Therefore, the function is increasing. Okay, So, continuing with a we have the derivative is a minus B over the B minus, C squared, which Okay, so see, here is the only variable. So this is just a constant A minus B times B minus C to the minus two. So if we take the second derivative, we can just leave this constant alone, which is positive, and then times negative two B minus C to the negative third times chain rule negative one. But this is good because this is just to a minus. B over B minus. C cubed and a minus. B is positive because it is bigger than B B is bigger than see, So this is positive. So this is hall positive. So the second report is positive. Therefore, the function is concave up, okay, And then in the second case, we actually don&#x27;t need to do any more work. We have the first derivative and the second derivative, so let&#x27;s just write them down. So if B is bigger than a C, so we still have first derivative is a minus B over B minus, C squared. But now this is negative. Cosby is bigger than a so the numerator is negative and then the same thing for the second derivative. Of course, that shows that in this case the function is decreasing and then second derivative is also negative. So it&#x27;s concave down and decreasing. Okay, so what sort of, uh, information can we draw? So I suppose that were we&#x27;re trying to sell product day. What does this mean? Of course, we want the probability function f to be greater well, in part a so part. A. So our product A has more, uh, unique features and as a function of C as a function of the number of common features, the functions increasing. So we want to make more common features because as a function of the number of common features, F is increasing probability that people will prefer people will prefer this Percy product day and then in the case that a has fewer unique features. The number of the probability is a function of the number of common features is decreasing, so we want fewer common features.

Bryan Meares · Answer

for questions. 87. We have a group of students who wants to perform an experiment to determine whether brand A or brand be deodorant last longer. One group member suggests the following design. Recruit 40 student volunteers 20 million, 20 female and separate the My gender basically give the males all one deodorant in the females, the other deodorant. And at the end of the school day, they should measure how well, the deodorants working on a scale between zero and 10, we&#x27;re gonna compare part a identify any flaws you see in the proposed design of this experiment? Well, because we&#x27;re separating the males and one group in the females in one group, we are having a little bit of confounding here based off gender. If we do find convincing evidence, we won&#x27;t really know if it&#x27;s based off the gender or the deodorant itself actually worked. So my answer for a confounding gender with veteran here. If convincing evidence is found, it won&#x27;t be clear if the reason was the gender of the participants or deodorant. Brand and B want to describe how we would design the experiment, and we&#x27;re gonna explain how this design addresses each of the problems identified in a. So we&#x27;re gonna design experiment here that fixes essentially the confounding element of gender. So the best product here because all humans are different. We all respond differently, um, to deodorant. Other things. Let&#x27;s do a match Paris&#x27;s on, in which each individual or receive both treatments. And we can compare how each treatment does for the individuals, So a better design would be a matched pairs design. We can have each volunteer flip a coin if heads, they were brand a on the left arm and be on the right arm. If tails the opposite is true at the end of the day, have each student compare the two brands. This eliminates confounding on gender.

Robin Corrigan · Answer

we need to state what type of test this is, given the options a through F, and we&#x27;re interested in the proportion of consumers who prefer and new laundry detergent over the leading competitors. So this is a question about one population and one proportion. It&#x27;s a single proportion, the consumers who prefer the new laundry over the leading laundry detergent, and so this is a single proportion test.

Problem

A survey by the National Institutes of Health ask…

Problem 50 Hard Difficulty

Answer

a. See histogram
b. No convincing evidence

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Daren S. Starnes, Daniel S. Yates, David S. Moore

The Practice of Statistics for AP*

Srikar K.

Boris M.

Monique R.

Lucas E.

Data And Variables

Data And Variables - Example 1

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a. See histogramb. No convincing evidence

The Practice of Statistics for AP*

Srikar K.

Boris M.

Monique R.

Lucas E.

Data And Variables

Data And Variables - Example 1

Daren S. Starnes, Daniel S. Yates, David S. MooreThe Practice of Statistics for AP*

Srikar K.

Boris M.

Monique R.

Lucas E.

a. See histogram
b. No convincing evidence

Daren S. Starnes, Daniel S. Yates, David S. Moore

The Practice of Statistics for AP*