# STAT1013: InClass Practice - A/B Test I

###### Suggested time: 25min

## Example

[5 Minutes statistics for clinical research - Definition of Hypotheses](https://youtu.be/IOmHkzTfbes)

## Problem 1.

In US, when a drug company wishes to introduce a new drug, they must first convince the [Food and Drug Administration](https://www.fda.gov/) (FDA) that it is a worthwhile addition to the current set of drugs. That is, they must be able to demonstrate that the new drug is better than current drugs, with “better” possibly referring to greater effectiveness, fewer side effects, or quicker response by the patient. There are risks in introducing a new drug — *is it worth it*?

Check this [video](https://youtu.be/kZCzN8FXt3I) for more details.

FDA guidelines say that the FDA doesn’t have to prove that a new drug is unsafe; rather, **the manufacturer of a new drug must prove that the drug is safe and effectiveness**.
This emphasis on safety has resulted in many drugs ultimately found to be dangerous not being made available in the United States; the anti–nausea drug thalidomide.

The hypothesis testing of FDA is summarized as:


Suppose you are a statistican conducting the FDA Phase I testing (safety) of a new drug. Note that the drug is approved as safe if the response metric is $E(X) = \mu < 0.001$, and you have observed 20 response metrics of some patients from a Phase I clinical trial.

1. What's the null and alternative hypothesis of the problem?
2. What's the Type I/II error of the problem.
3. Briefly specify steps (not technical detail) of conducting a hypothesis testing of Phase I of this new drug.
4. Please state in which situations you believe rejecting the null hypothesis is appropriate.
5. Specify all possible conclusions of this testing.

## Problem 2 (**for fun**).
Define a two-player game:

1. There are one $10 and two $1 bills in a black box.
2. Each of player take (but can not see) one bill, and show his/her own bill to the other player. That is, the player knows the bill of the other player, but not his/her own bill.
3. Both player are perfectly logical. If a player know his/her own bill, then he/she will take the bill.

:::info
Q: Itemize all possible outcomes of the game.
:::

## The Blue-Eyed Islanders (for fun;optional)

On an otherwise deserted and isolated island, 200 perfect logicians are stranded. The islanders are perfectly logical in every decision they make, and they will not do anything unless they are absolutely certain of the outcome. However, they cannot communicate with each other. They are forbidden from speaking with one another, or signing, or writing messages in the sand, else they be shot dead by the captain of a mysterious ship that visits the island each night.

Of the 200 islanders, 100 have blue eyes, and 100 have brown eyes. However, no individual knows what color their own eyes are. There are no reflective surfaces on the island for the inhabitants to see a reflection of their own eyes. They can each see the 199 other islanders and their eye colors, but any given individual does not know if his or her own eyes are brown, blue, or perhaps another color entirely. And remember, they cannot communicate with each other in any way under penalty of death.

Each night, when the captain of the ship comes, the islanders have a chance to leave the barren and desolate spit of land they have been marooned on. If an islander tells the captain the color of his or her own eyes, they may board the ship and leave. If they get it wrong, they will be shot dead.

Now, there is one more person on the island: the guru, who the islanders know to always tell the truth. The guru has green eyes. One day, she stands up before all 200 islanders and says:

I see a person with blue eyes.

:::info
Q: Who leaves the island? And when do they leave?