Assignment
1. Fun with estimators The usual estimator for a population proportion, p, is pˆ = Y1+···+Yn-1 + Yn , where Y is Bernoulli and n is the sample size. An alternative estimator for p is p˜ =Y1+...+Yn+1+Yn+2.
(a) Is pˆ unbiased? Is p˜ unbiased? If "yes," show why. If "no," show the amount of bias (which may depend on sample size or p).
(b) Same as (a), but now check whether they're asymptotically unbiased.
(c) Is pˆ consistent? Is p˜ consistent? Again, show why or why not.
(d) Which estimator is more efficient, pˆ or p˜. Does it depend on sample size or p?
(e) Does what you've found above conflict with pˆ being BLUE? Why or why not?
(f) You want to estimate the proportion of students who take public transit to the U. You can take a random sample of students. Obviously, you don't know p. Are there any circumstances (sample size and p) under which you would be better off using p˜ instead of pˆ to estimate p. Explain, describing the circumstances (if any) and the notion of "better" you're using to choose.
2. What's the best way to estimate a product? Two survey analysts - for lack of my imagination just call them A and B - are having a dispute. They have conducted a survey aimed at estimating the amount of time students spend in-commute to and from the U. The survey is simple: A random sample of commuting students are asked (1) how many commutes to campus did you make last week, and (2) how many minutes, on average, is your commute? Analyst A proposes the following estimator: Calculate the average number of commutes, call it C, and the average (of the average) time in minutes per commute, call it T, then multiply: C×T. B proposes the alternative: First multiply each subject's reported number of commutes and time per commute, then compute the average of these products. Who is right, which estimator is better? In what sense is that one better? Are there circumstances in which the estimators are equally good? Explain. Are these circumstances, if any, likely to hold here?
3. Test scores Suppose a new randomized test is given to 100 randomly selected third-grade students in New Jersey. The sample average score Y on the test is 58 points, and the sample standard deviation, sY , is 8 points.
(a) The authors plan to administer the test to all third-grade students iin New Jersey. Construct a 95% confidence interval for the mean score of all New Jersey third graders.
(b) Suppose the same test is given to 200 randomly selected third graders from Iowa, producing a sample average of 62 points and sample standard deviation of 11 points. Construct a 90% confidence interval for the difference in mean scores between Iowa and New Jersey.
(c) Can you conclude with a high degree of confidence that the population means for Iowa and New Jersey students are different? (What is the standard error of the difference in the two sample means? What is the p-value of the test of no difference in means versus some difference?)
4. Does something become more valuable to you simply by virtue of your owning it? A statistical test of the endowment effect. Use R to answer the following questions. Submit your answers on paper and the R code you used to get those answers to Canvas. Part (a) doesn't require R so your answers to (a) can be submitted on paper. Also, in part (a) keep in mind that what you're trying to do is work out what you would expect to see in the data if there were no endowment effect. When you analyze the data you need to know what you would expected to see if the hypothesis (that there's an endowment effect) is false. In this case, as you'll show, if the hypothesis is false we should see about-not exactly, this is a sample-50% of people trade their item.
A consumer is given the chance to buy a baseball card for $1, but he declines the trade. If the consumer is now given the baseball card, will he be willing to sell it for $1? Standard consumer theory suggests yes, but behavioral economists have found that "ownership" tends to increase the value of goods to consumers. That is, the consumer may hold out for some amount more than $1 (for example, $1.20) when selling the card, even though he was willing to pay only some amount less than $1 (for example, $0.88) when buying it. Behavioral economists call this phenomenon the "endowment effect." John List investigated the endowment effect in a randomized experiment involving sports memorabilia traders at a sports-card show. Traders were randomly given one of two sports collectibles, say good A or good B, that had approximately equal market value. Those receiving good A were then given the option of trading good A for good B with the experimenter; those receiving good B were given the option of trading good B for good A with the experimenter. Data from the experiment can be found on Canvas in the file sportscards.csv and a description of the data in sportscards_description.pdf.
(a) i. Suppose that, absent any endowment effect, all the subjects prefer good A to good B. What fraction of the experiment's subjects would you expect to trade the good that they were given for the other good? (Hint: Because of random assignments, approximately 50% of the subjects received good A and 50% percent received good B.) ii. Suppose that, absent any endowment effect, 50% of the subjects prefer good A to good B, and the other 50% prefer good B to good A. What fraction of the subjects would you expect to trade the good that they were given for the other good?
iii. Suppose that, absent any endowment effect, X% of the subjects prefer good A to good B, and the other (100-X)% prefer good B to good A. Show that you would expect 50% of the subjects to trade the good that they were given for the other good.
(b) Using the sports-card data, what fraction of the subjects traded the good they were given? Is the fraction significantly different from 50%? Is there evidence of an endowment effect?
(c) Some have argued that the endowment effect may be present but that it is likely to disappear as traders gain more trading experience. Half of the experimental subjects were dealers, and the other half were non dealers. Dealers have more experience than non dealers. Repeat (b) for dealers and non dealers. Is there a significant difference in their behavior? Is the evidence consistent with the hypothesis that the endowment effect disappears as traders gain more experience?