Denomination Effect A trial was conducted with 75 women in Chi...

Question

Denomination Effect A trial was conducted with 75 women in China given a 100-yuan bill, while another 75 women in China were given 100 yuan in the form of smaller bills (a 50-yuan bill plus two 20-yuan bills plus two 5-yuan bills). Among those given the single bill, 60 spent some or all of the money. Among those given the smaller bills, 68 spent some or all of the money (based on data from “The Denomination Effect,” by Raghubir and Srivastava, Journal of Consumer Research, Vol. 36). We want to use a 0.05 significance level to test the claim that when given a single large bill, a smaller proportion of women in China spend some or all of the money when compared to the proportion of women in China given the same amount in smaller bills.

a. Test the claim using a hypothesis test.

a. Test the claim using a hypothesis test.

Accepted Answer

Researchers surveyed a simple random sample of 120 college students in Brazil. 60 students were given a single 100 real note, while the other 60 received the same amount in smaller denominations, being 520 real notes. Among the single note group, 48 students spent some or all of the money. In the small denomination group, 55 students spent some or all of the money. Using a hypothesis test at the 0.05 significance level, Tesla claimed that the fewer students spent the money when given a single large note. Now, to solve this, let's first set up. Our assumptions. We need to have a simple random sample, which we do, it gives it to us at the top. Now we need to have independent samples from each other. We have the 100 real note sample and the 20 real note sample. So, these are independent samples, which confirms our condition. We also need to check If we have a certain number of successes and failures. In particular, for our proportions, we need to see if our sample size multiplied by our proportion is greater than or equals to 5. We also need to check if our sample size multiplied by the complement or 1 minus the portion is also greater equals to 5. So, we can then see N1 multiplied. By Phat one. Which will be 60, multiplied by our first proportion. 48 divided by 60. Which just gives us 48. And 1, multiplied by 1 minus P1. Gives us 60 multiplied by 1 minus 48 divided by 60. Which gives you 12. We'll also check our other sample. In 2. Multiplied I had to Which is 60, multiplied by 55 divided by 60. This gives us 55. And in 2 multiplied by 1 minus Pha 2. Which is 60, multiplied by 1 minus 55 divided by 60, or just 5. All of these are greater than or equals to 5, so we can move on with our normal approximation assumption. Now let's set up our hypothesis tests. We have our null hypothesis and our alternative hypothesis. In this case, our null hypothesis is the is the condition we already know. In this case, We should know that our first proportion is greater than equals to our second proportion. So we will say P1 is greater than equals to P2. Now, we're testing the claim that fewer students spent money when given the single large note. So we're testing that if our first proportion spent less than our second proportion, P1, is less than P2. Now that we have our hypotheses set up. We can go ahead and move on finding our test statistic. To find this, we'll first find our pooled. Sample proportion This is given By X1 plus X2, divided by N1 plus N2. Or in other words, 48 plus 55. Divided by 60 plus 60. We get 103 divided by 120. Which rounds to 0.8583. We can also find the compliment. By saying Q equals 1 minus the proportion. That's 1 minus 0.85 83. Which is 0.1417. Now, we'll go ahead and calculate our test statistic. This formula is given by. See? Equals Phat 1 minus Phat 2. Minus P1 minus P2. All divided By the square root Of P bar multiplied by Q bar. Divided by N1 plus Par multiplied by Q bar, divided by N2. Let's plug in all of our values. We have Phat one, was from earlier, is 0.8. And PAT 2, which is 0.91 67, we get these by finding our proportions, 48 divided by 60 and 55 divided by 60. This is divided by 0.8583. Multiply By 0.1417. Divided by 60 Plus 0.8583, multiplied by 0.1417 divided by 60. We can then simplify this to get approximately 1.833. Note that P1 minus P2 is 0 based on what we found in our hypothesis as well. So I'll put a 0 there to note that. Now, we just need to find our critical value. R Z critical is given by the Z table. And so, if we look at our test again, it says we want. A 0.05 significance level test. The Z critical value for 0.05. is given by -1.6 or 5. We can find this on the Z table as mentioned before. Now we notice our value is -1.833. Z is less than Z critical in our case. Because 1.833 is less than -1.645. Because our Z is less than Z critical, based on the one tail tests, or the left tail test inarcus. We can reject. Our null hypothesis. We then claim that there is Sufficient Evidence To support Our claim. In this case, our claim is that there's a small proportion of students that spent money using the single real notes. So we'll say there is sufficient evidence to support. That A smaller Proportion Of students Spent Money With The 100 real. Notes. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

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Key Concepts

Hypothesis Testing

Significance Level

Proportions and Comparison

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