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Credit cards and incentives A bank wants to know which of two incentive plans will most increase the use of its credit cards. It offers each incentive to a group of...

Question

Credit cards and incentives A bank wants to know which of two incentive plans will most increase the use of its credit cards. It offers each incentive to a group of current credit card customers, determined atrandom, and compares the amount charged during the following six months.(a) Is this a problem about comparing means or comparing proportions? Explain.(b) What type of study design is being used to produce data?

Credit cards and incentives A bank wants to know which of two incentive plans will most increase the use of its credit cards. It offers each incentive to a group of current credit card customers, determined at random, and compares the amount charged during the following six months. (a) Is this a problem about comparing means or comparing proportions? Explain. (b) What type of study design is being used to produce data?



Answers

Credit cards and incentives A bank wants to know which of two incentive plans will most increase the use of its credit cards. It offers each incentive to a group of current credit card customers, determined at
random, and compares the amount charged during the following six months.
(a) Is this a problem about comparing means or comparing proportions? Explain.
(b) What type of study design is being used to produce data?

Hi, everyone. Today we're starting chapter 34 with question one where we're being asked to explain how each of the following developments would affect the supply of money, the demand for money and the interest rate. And we should illustrate our answers with diagrams. All right, so first of all, we don't determine what is the right kind of background we're gonna be using. Since our questions deal with a money market, it makes sense to use a simple supply and demand graph for the money market. Namely on the X axis. We have a quantity of money. Em on the Y axis. We have interest rates. Are the money supply curve will always be very cool because it's fixed by the central bank exogenous Lee and the money demand curve will be downward. Sloping. Now, just a quick reminder. Why's the money Domenic Irv downward sloping according to the theory of requested preference that we'd learned in chapter 34 the interest rate is the opportunity cost of holding money that is one. You hold wealth as cash in your wallet instead of as an interest bearing bonds he'd lose dangerous. You could have earned as a result and increasing the interest rate raises the cost of holding money and consequently reduces according to find demanded and vice versa. So we see there is a negative relation between the interest on the interest rates in money holdings, and hence the card will be Dad would stuff. All right, So part of one asks what happens when the feds bond traders buy bonds and open open market operations? Well, this is not a very difficult question, because we know that when the feds bond traders buy bonds, they increase the supply of money in the economy. So the money supply curve will move from M s one m iss to those shift to the right. And since this die, sex and downward sloping on domenica of Lower Point, are you a liberal interest rate would be lower women from our warrant, Artoo. And of course, this makes sense. Since the buying of bonds in open market operations in checks minded the economy, money becomes more abundant, and hence the interest they're going to be baring must be lower. Okay, Part B asks what happens when an increasing credit card availability reduces the amount of cash. People want hope This is even more straightforward because we've been given that people want to hold less money. So what does this mean? That the demand for money decreases. So the one demanded Carver will shift to the left MD one coast of empty, too. And hence, once again, the equilibrium interest is gonna drop from our warn. You are too parte si says what happens when the Federal Reserve reduces bank's reserve re part mints? Okay, well, let's think about it for a second. One of the many tools of the Fed is changing or setting the targets for the banks with the minimum banksreserve requirements. So if the feds says that now that banks needs to have 10% instead of 20% in reserve requirements, one of the bags going to do, you're going to be able to lend more money. Yeah, through loans. And this will definitely increase the one supply. So the months of, like her ships from M. S one m isu and once again think a liberal interest rate reduces toward two. This is very similar to, ah, to our first case party. All right, Part four says how scolds decide to hold more money to use for holiday shopping once again. Here it is. The answer is very straightforward, because we've been given that households want more money, so we know that the money Domenica will sift to the right from MD one empty too. And since money becomes more wanted in this economy, think about it as a more scars commodity. The interest the interest interest rate will have to increase from our one part two. Finally, we're being asked to determine what happens when a wave of optimism boosting but business investment and expands angry demand once again similar to party. What's happening here is that business become very optimistic about the future they wanted. Invest more than I want to take more loans, and this will increase the demand for money since agreement expense. So the money demand Carvel shift from MD want empty, too. And once again, since money Demet money because I become the more scars commodity more one calamity in the economy, the interest rate will rise from our want art, too

In this problem were given the probabilities of customers at a gas station paying with credit card, debit card or cash. And for part a were asked that among the next 100 customers, what are the mean and variance of the number who pay with a debit debit card? If we call X the number who pay with debit, we can say there are 100 customers, and the probability of paying with a debit card is given here. So thats 0.2, and we can also make the argument that X is a binomial random variable based on 100 trials and probability of success of 0.2, because we believe that all of the customers are choosing credit, debit or cash independent of one another. And we've defined our outcome as either a success, which is pay with debit or a failure. So, given this, then the expected number of successes for a binomial random variable is given times by end Times P, which is 100 times 0.2, which gives US 20 and the variance of X is given by N times p Times Q, which is 100 times 0.2 times 0.8 and this comes out to 16 and were asked to explain our reasoning. So that's what we did by justifying that it was a binomial random variable. We were able to frame the question as a binomial by ensuring that there is only two possible outcomes a success or a failure. And then, for Part B, were asked to answer Part A for the number among 100 who don't pay with cash. So if 30% pay with cash, that means 70% dont pay with cash. So let's say P a 0.7. And in this case we're calling X a success. We're seeing X is the number of successes, and we're saying X is the number of people who do not pay with cash. So successes somebody not paying with cash and N is still 100. So now we can say that X is a binomial random variable based on 100 and 0.7. So now the expected number of people who do not pay with cash is an Times P, which comes out to 70 and the variance of X end times P. Times Q. And that is equal to 21

In this problem, a series of probabilities is given to you. You were given that the probability that a consumer uses a plastic card when making a purchase is 0.37 The probability that a consumer is 18 to 20 figure four years old, given that they use a plastic card equals 40.19 The probability that a consumer is 24 years old or older, given that they use a plastic card equals 240.81 and that 14% of the consumer population is 18 to 24 years old. Heart a asks. Given that the consumer is 18 to 24 years old, what is the probability that they use a plastic card? This is the probability of using a plastic card, given that the consumer is 18 to 24 years old, which equals the joint probability between using a plastic card and being 18 to 24 years old, divided by the marginal probability of being 18 to 24 years old. The joint probability is not given in the problem, and this must be calculated. To do this, we can take the information given above, so the probability the consumer is 18 to 24 years old, given that they use a plastic card which equals 0.19 and said that equals to the joint probability divided by the marginal probability of using a plastic card. This is 0.19 equal to the joint probability divided by 0.37 you. Therefore, it can multiply 0.19 by 0.37 to get the joint probability between being 18 to 24 years old and using a plastic card. This calculation gives a joint probability of 0.703 You can plug this in to the original equation to get 0.703 divided by 0.14 for a probability of 0.50 to 1 part B. Ask. Given that the consumer is 25 years old and older, what is the probability of using a plastic card? This is the probability of using a plastic card, given that there 24 years older, older, which is equal to the joint probability between using a plastic card and being 24 years old and older, divided by the marginal probability of being 24 years old and older. Like the previous problem, we do not have the joint probability between using a plastic card and being over 24. We therefore calculated using the numbers given in the problem. We can take the probability of being 24 years old and older, given that they used a plastic card which he goes 240.81 and said that equal to the joint probability divided by the marginal probability of being plastic, this is 0.81 equal to the joint probability divided by 0.37 To get the joint probability, we therefore multiply 0.81 point 37 to give a drink probability of 370.2997 You can put this into the previous equation to get 0.2997 divided by the probability of being 24 years old and older, which is one minus. The probability of being 18 to 24 so one minus 10.1 for to give a probability of 0.3485 Part C Asked to interpret the probabilities from Part A and B hurt A was the probability using a plastic card given that you're 18 to 24 years old and equaled 0.50 to 1 The interpretation is that 50.21% of the population of 18 24 year olds uses a plastic card when making a purchase. Part two was the probability of using a plastic card given that you were 24 years old or older and that equalled 0.3485 The interpretation is that 34.85% of the population of 24 year olds and older use a plastic card when making a purchase, Per Di asked. Should credit card companies such as Visa MasterCard, etcetera make plastic cards available to the 18 to 24 year old age group for these consumers have had time to establish a credit history. The answer is yes. As shown in the previous part, roughly half of this age group uses a plastic card when making a purchase, as opposed to only 1/3 of the older population using a plastic card. This helps these younger consumers to establish credit history, however, because of their lack of history restrictions such as withdrawing limit so that they can't spend more than, say, 1000 or $2000 a month can be set so that the risk to the credit card companies is limited

And this problem. We're trying to see if the method of payment is independent of age or not. Um so our null hypothesis is that, um the row and column variables are independent, and our alternative hypothesis is that the row and column variables are not independent. So in this situation, it would be that the age group and the payment method are not independent. So we're given this data set, and these air are observed values. And I went ahead and calculated the totals for our rose and for a rose and the totals for our columns and then our grand total. So we have this, and what we need to do to test our hypotheses is come up with a chi squared value in the chi squared. Value is the observed frequency for each row and column combination minus the expected frequency for each Roman column combination squared, divided by the expected frequency for each Roman column combination. So what? I am What do I mean by Rohan Column combination. So we have two rows and, uh, four columns, so this would be the combination for plastic in 18 to 24. This would be the combination for plastic in 25 to 34 so each of our cells represent a row and column combination. What we need to do is come up with the expected values and now are expected values for each Roman column combination is going to be the row total. So the end and that, um column times the n in that row divided by the total. Okay, so for our row of plastic, we have a total of 1 11 This would be 1 11 times are over column total of 42 a 42 divided by our grand total of 300. So with this, we get a value of Let me make sure I'm saying this right, 15.54 15.54 And we're going to do this for each of ourselves. So this is going to be 26.6646 This is 23.31 39.69 25.53 43.47 and 46.62 and 79.38 Now to come up with our chi squared, we need to come up with the difference between each cell and square it divided by our, um, expected frequency. So what I'm going to do is come up with another table that does this math for us and holds thes values. So for plastic and 18 to 24 the difference of the square difference over the expected frequency is approximately 1.92 for plastic and 25 to 34. It's 0.58 for the plastic and 35 to 54 combination, 0.850 point zero nine. 45. Plus it's 2.42 and these air rounded valued, by the way. So I'm just gonna fill out the rest of this table. So for our chi squared value, we need to come up with some of all the values in our cell, right in our table. In our in this difference table, S. O. R. Ky squared is going to be equal to 1.92 plus 0.58 plus 0.9 Continue. Continue. Continue until you get to plus 0.5 plus 1.42 and we get a chi squared value of seven. Approximately seven point 95 now, given that Sark, I squared value. We need to come up with a degrees of freedom in order to find our P value. This is the formula for the degrees of freedom. The number of rows minus one times a number of columns minus one. So we have two rows minus one, and then we have four columns minus one. So this is equal to one. This is equal to three. So one times three is three. So now what? We have to find a P value, given that our chi square to 7.95 and our degrees of freedom is three. So, me yet that our P value is, um, between 0.0 point 0 to 5. So the range is 0.252 point 05 And now we are testing this against an Alfa of 0.5 So because our he is in between 0.0 to 5 and 0.5 which is less than or equal to our Alfa, which equals 0.5 we reject the gnome. So what does that mean? It means that age and payment method are are not independent of each other. Um, so they're not independent. So what does this mean? For part? Be in part B were asked to find, um what observations could be make about the different age groups and their use of plastic or cash. So, in part B, you can come up with this conclusion or this statement, Um that, ah, the payment method for people aged 18 to 24 that used plastic. So payment for people aged 18 to 24 who use plastic is equal to 21 over 42 times 100 which is equal to 50%. And where am I getting these numbers from? If we go to our observed, um, table, I have 21 here, and that's the number of people that use plastic and who are aged 18 24. And that is over our total number of people aged 18 to 24. That's 42. So that's 50%. And the percentage of people that use, um so mission right pay instead of payment that are aged between 25 to 34 use plastic is equal to 27 over 69 times 100 which is equal to, uh, 27 over 63 sorry, 63 times 100 is equal to 42.86%. The payment method, given that they're 35 to 40 four and use plastic is equal to 27 over 69 times 100 which is equal to 39.13%. And payment method. Given that they are 45 and over and use plastic is equal to 36 divided by 90 times 100 illegal tow 40%. So we see that, um, the youngest age group used plastic to make purchases more frequently than any of the other age groups. So our conclusion for Beacon be, um, youngest age group uses cat ah, plastic more frequently than other age groups. And now we have Thio figure out. If, ah, companies such as Visa, MasterCard Discover can use this. So what they know is that because these aren't independent for their marketing strategy, they shouldn't just give out one ad for all age groups thinking that they can collect everybody. They should have targeted ads based that could be one possible thing. They could do targeted ads at each a troop, and they should also focus more on the older age groups, Um, and get them up to that 50% threshold, that 50% threshold before they try to get more people from the younger groups or any something like that, UM targeted ads and try to get older age group. Or maybe they can think that this is a some cost and say it's not worth looking at the older age groups because they are not interested in. They're not interested in plastic. It'll it they're only interested in cash, so we should just focus on recruiting more young people to use.


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