BASIC CONCEPTS OF HYPOTHESIS TESTING

BASIC CONCEPTS OF HYPOTHESIS TESTING

Designed by Engin A. Sungur

SCENARIO


In every issue that we face with we have some prior belief/opinion. When we receive enough information/evidence we revisit our prior belief/opinion. Based on the strength of information/evidence we may collect, we either change of prior belief/opinion or preserve our prior opinion by saying that "there is not enough evidence/information provided to me to change my prior belief/opinion.

QUESTION

Now, we will play a game. We will toss a coin, if it turns out a head, let us say I will win, if it turns out a tail you will win. Is this a "fair" game?

OBJECTIVES

To understand the logic behind the test of hypothesis, to see how people reject hypothesis upon observing data that are unlikely to arise if the hypothesis is true. And, to grasp the meaning of P-value in the test of hypothesis.

ACTIVITY

a. To warm-up click on the "toss a coin" button on the left window to toss the coin. Toss the coin 10 times. If the coin goes outside of your screen click on the "refresh left side" button and continue.


Now, you met with the coin that we are going to play our game.

b. Click on the "play" button, to start the game.


Click on the "toss the coin" button on the left window and fill out the rows of the following table at the end of each toss by recording the result(head or tail), cumulative number of heads, proportion of heads(cumulative heads/total number of tosses). Be sure to record your reaction after each repetition on the last column by referring what conclusions you feel you can draw regarding the question "Is the game fair?" and how sure you feel about the answer. (If the coin goes outside of your screen click on the "refresh left side" button and continue.)


Toss #
Result/

Outcome
Cumulative Heads
Total Number of Tosses
Proportion of Heads
Thoughts and Remarks
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11

c. At the beginning of the game, the probability of you winning the game, that is probability of getting a tail was 0.5. Under the assumption that the game was a fair game fill out the following table:

Toss #Outcome Probability of Observing this Outcome
1H0.5
2HH(0.5)x(0.5)=0.25
3HHH
4HHHH
5HHHHH
6HHHHHH
7HHHHHHH
8HHHHHHHH
9HHHHHHHHH
10HHHHHHHHHH
11HHHHHHHHHHH

d. By referring to part b, after how many tosses you started to think that the game is not a fair game, that is the probability of getting a tail is not 0.5?

From your response to part c, what is the probability of observing that many heads, if the coin was fair?

Note that the last column of the table given in part c, is called P-value of the test in test of hypothesis set up. The P-value of the test is the probability of getting data at least as extreme as the data you obtained, under the assumption that the game was fair.

EXTENSION

Let us play another game. This time we will use a die. I win on odd numbers (1, 3, 5); you win on even numbers (2, 4, 6).

Click on the "Play" button to start the game.


Click on the "roll a die" button on the left window to play the game. Fill out the rows of the following table at the end of each roll by recording the result(odd or even), cumulative number of odd numbers, proportion of odd numbers(cumulative number of odd numbers/total number of rolls). Be sure to record your reaction after each repetition on the last column by referring what conclusions you feel you can draw regarding the question "Is the game fair?" and how sure you feel about the answer. (If the die goes outside of your screen click on the "refresh left side" button and continue.)




Roll #
Result/

Outcome
Cumulative Number of Odd Numbers
Total Number of Rolls
Proportion of Odd Numbers
Thoughts and Remarks
1
1
2
2
3
3
4
4
5
5
6
6
7
7
8
8
9
9
10
10
11
11

At the beginning of the game, the probability of you winning the game, that is probability of getting a tail was 0.5. Under the assumption that the game was a fair game fill out the following table (E stands for an even number):

Toss #Outcome Probability of Observing This Outcome
1E0.5
2EE(0.5)x(0.5)=0.25
3EEE
4EEEE
5EEEEE
6EEEEEE
7EEEEEEE
8EEEEEEEE
9EEEEEEEEE
10EEEEEEEEEE
11EEEEEEEEEEE

After how many rolls you started to think that the game is not a fair game, that is the probability of getting an odd number is not 0.5?

Is this different than your answer to part d in the main activity? If yes what is the reason?

ASSESSMENT

Write down what you learned in this activity about the test of hypothesis. Also, define P-value in your own terms.