Question

	Matching Grants		Sum:
Education charities	$370,272	$260,071
Religious charities	$932,207	$300,122
Sum:

1. (2 pts) Practicing conditional distributions and experimental vocabulary:

Note: I have used a random number generator to select student names for this question, to select whose hypothesis to make be correct, and to select which type of table to have you actually create.

Imagine that we have run an experiment where we have taken a list of people known to donate small amounts to charities and randomly assigned each person to one of four possible fundraising campaigns. We have generated a simple random sample; every person has an equal chance of being assigned to each treatment.

Campaign 1 raises money for an education charity, and features a matching grant. (Large donors have pledged to match every dollar raised from small donors, so a $10 grant leads to $20 in money for the charity. Note that Table 1 includes only the funds from the small donors, not the matching gifts.)

Campaign 2 raises money for an educational charity, but features a lottery. Every dollar donated gives the recipient a chance to win an hour of free statistics lecturing from Professor Weinberg.

Campaign 3 raises money for a religious charity, with a matching grant.

Campaign 4 raises money for a religious charity, with a lottery.

1. (0.5 points)
   1. What do we call the list of donors that we used to randomly assign people to treatments?
   2. What are the two factors in this experiment?
   3. What are the four levels in this experiment? (Hint: each factor has two levels.)

Table 1: Dollars Raised from Small Donors by Type of Charity and Fundraising Strategy

	Matching Grants	No Matching Grants
Education charities	$370,272	$260,071
Religious charities	$932,207	$300,122

Source: totally made-up data

Herman, Angela, and Robert are arguing about whether matching grants are more helpful for certain types of charities.

Herman argues that matching grants are particularly helpful for religious charities, because religious charities raise 76% of their funds from matching grants, whereas education charities raise only 59% of their funds from matching grants.

2. (0.25 points) Is Herman’s argument based on a conditional distribution or a joint distribution? How can you tell? If it is a conditional distribution, what is the factor being conditioned on?

Angela argues that matching grants cannot be more important for religious charities, because religious charities raise more money from lotteries than educational charities do. Angela points out that 16% of all funds come from religious campaigns with lotteries, compared to only 14% for educational campaigns with lotteries.

3. (0.25 pts) Is Angela’s argument based on a conditional distribution or a joint distribution? How can you tell? If it is a conditional distribution, what is the factor being conditioned on?

Robert agrees with Angela, but argues that she used the wrong evidence. He says the important thing here is that, for the lottery campaigns, 46% of funds were raised for education charities and 54% of funds were raised for religious charities.

4. (0.25 pts) Is Robert’s argument based on a conditional distribution or a joint distribution? How can you tell? If it is a conditional distribution, what is the factor being conditioned on?

5. (0.25 pts) Whose argument is the most persuasive? Explain.

(0.5 pts) Construct a conditional distribution table, conditioned on type of charity. (I chose this table randomly; my asking for it is not a clue about which table is most relevant for this question. I simply didn’t want to make you create all three types.

Answer 1

1) The list of donors we used to randomly assign people to treatments are called experimental units.

2) The two factors of this experiment are education charities and religious charities.

3) The four levels are matching grants with education charity,no matching grants with education charity, matching grants with religious charity and no matching grants with religious charity.

4)Firstly the complete table is :

	Matching Grants	No Matching Grants	Sum:
Education charities	$370,272	$260,071	$630,343
Religious charities	$932,207	$300,122	$1,232,329
Sum:	$1,302,479	$560,193

Herman's argument is based on conditional distribution as we can see that when herman says that religious charities raise 76% of their funds from matching grants, he means $\frac{932207}{\sum religious charities}=\frac{932207}{1232329}$ which implies that he has conditioned Matching Grants when religious charities is given i.e. P(Matching Grants I Religious charities).

59% of their funds from matching grants, he means $\frac{370272}{\sum Educationcharities}=\frac{370272}{630343}$ which implies that he has conditioned Matching Grants when education charities is given i.e. P(Matching Grants I Education charities).

In joint distribution, the probability must have been P(Education charities,Matching Grants) = $\frac{370272}{Total }$ = $\frac{370272}{1862672}$

5) Angela's argument is based on joint distribution as we can see that Angela says that 16% of all funds come from religious campaigns with lotteries, she means $\frac{300122}{Total} = \frac{300122}{1862672}$ which implies that she is talking about the joint distribution.

Angela's argument is based on joint distribution as we can see that Angela says that 14% of all funds come from educational campaigns with lotteries, she means $\frac{260071}{Total} = \frac{300122}{1862672}$ which implies that she is talking about the joint distribution.

6) Robert's argument is based on conditional distribution as we can see that Robert says that for the lottery campaigns, 46% of funds were raised for education charities, he means $\frac{260071}{\sum lotteries}=\frac{932207}{560193}$ which implies that he has conditioned Education charities when lottery is given i.e. P( Education charities I Lotteries).

And for 54% of funds were raised for religious charities, he means $\frac{300122}{\sum lotteries}=\frac{300122}{560193}$ which implies that he has conditioned religious charities when lottery is given i.e. P( Religious charities I Lotteries).

7) Angela's arguments is most persuasive as she is using joint probability and not conditioning a factor.

8) Conditional distribution conditioned on type of charity:

	Matching Grants	No Matching Grants	Sum:
Education charities	0.587	0.412	$630,343
Religious charities	0.756	0.243	$1,232,329