Hypothesis Testing: Dichotomous Variables

1. Some Definitions
2. The Coin Problem
   1. Constructing the Binomial
   2. Testing Hypotheses
      1. State the Hypotheses
      2. Choose a Significance Level
      3. Assume the Null Hypothesis
      4. Compute the Probability of the Observed Outcome - [Minitab]
      5. Make a Decision
   3. Other Relevant Issues
   1. Directional vs. Nondirectional Hypotheses
   2. Accepting, Asserting, and Rejecting
   3. Probabilistic Nature of Science

Homework

I. Some Definitions

1. Probability
   Refers to the liklihood that an event will occur. Ranges from 0 to 1. An event with a probability of zero will not happen and an event with a probability of one will definitely happen.
   

   For discrete variables:

   P(A)=

   # outcomes favoring A   Total # possible outcomes

   Ex. Probability of drawing an ace from a deck of cards is 4/52.

   For continuous variables:

   P=	Area under a portion of the curve     Total area under the curve

   Since the total area under the curve is 1, we define P as the proportion of area under the curve. Ex. The probability of choosing a random person with an IQ between 85 and 115 is .6826

2. Dichotomous Variable
   A discrete categorical variable with two possible values. Exs. head/tail, pass/fail, male/female, do or don't do something, have a positive or negative opinion about something.

3. Sampling Distribution
   A probability distribution of the possible values of some sample statistic which would occur if we were to draw all possible samples of a fixed size from a given population. Note a sampling distribution tells us two things:
   1. All possible values of a statistic.
   2. Their probabilities of occurrence.

4. Binomial Distribution
   A sampling distribution of a dichotomous variable. Since a dichotomous variable is the simplest type of variable, a binomial distribution is thus the simplest type of sampling distribution.

II. The Coin Problem

This problem will help illustrate the terms defined above as well as the basic logic for all of hypothesis testing. Let’s say you use a coin to help you make decisions. 'Heads' means you study and 'tails' means you socialize. However, you feel that you are getting the raw end of the deal and believe the coin is biased. Thus, the research question asks whether the coin is biased. Let us examine the method for answering this question. Before we are ready to start flipping the coin a bunch of times and test the hypothesis, we need to get a deeper understanding of the concept of a Binomial Distribution that we defined above.

1. Constructing the Binomial Distribution

   Explanation

   First let us consider what can happen if we toss a coin once (i.e., N=1). The possible outcomes are:

   A head (H) or a tail (T).

   In summary, the binomial distribution for N=1 is:
   
   

   Possible outcomes	H1T0	H0T1
   Probability	1/2	1/2

   Next let us consider what can happen if we toss a coin twice (i.e., N=2). The possible outcomes are:

   2 heads (H²), a head and a tail (HT), and 2 tails (T²).

   Note, however, that the head and a tail can occur in 2 ways (HT or TH). Thus, the possible outcomes are:

   2 heads (1H²), a head and a tail (2H¹T¹), and 2 tails (1T²).

   The bold numbers are called coefficients and indicate the number of different ways an outcome can occur. Note, that the sum of the coefficients is equal to all possible outcomes (i.e., 1 + 2 + 1 = 4).

   In summary, the binomial distribution for N=2 is:
   
   

   Possible outcomes	H2T0	2H1T1	H0T2
   Probability	1/4	2/4	1/4

   
   Notes:
   1. As with all sampling distributions, all possible values of a statistic and their probabilities of occurrence are given.
   2. N+1 = the number of different possible outcomes. In the present case, there are 3 different possible outcomes (i.e., H²T⁰, H¹T¹, H⁰T²).
   3. 2^N = åcoefficients = the total number of all possible outcomes. In the present case, there are 4 possible outcomes (i.e., HH, HT, TH, TT).
   4. The probabilities for all outcomes sum to 1.

   Now consider tossing the coin three times (i.e., N=3):
   
   

   Different possible outcomes	H3T0	H2T1	H1T2	H0T3
   All possible outcomes	HHH	HHT HTH THH	HTT THT TTH	TTT
   Coefficients	1	3	3	1

   Thus, the binomial distribution for N=3 is:
   
   

   Possible outcomes	H3T0	H2T1	H1T2	H0T3
   Probability	1/8	3/8	3/8	1/8

   As you might guess, constructing a binomial when N>3 gets to be difficult and thus there are more efficient methods.

   A couple of methods to generate coefficients more quickly:

   1. Expand the Binomial. Exs:

      (a + b)2 = (a + b)(a + b)
               = a2 + 2ab + b2
      
      (a + b)3 = (a + b)(a + b)(a + b)
               = (a + b)(a2 + 2ab + b2)
               = a3 + 2a2b + ab2 + a2b + 2ab2 + b3
               = a3 + 3a2b + 3ab2 + b3
      

      Note that the exponent equals the number of tosses. However, this, too, is a tedious technique.
      
      
   2. Pascal’s Triangle
      
      A trick with numbers that generates the coefficients.

      N	Coefficients	N+1	2n
      1	1 1	2	2
      2	1 2 1	3	4
      3	1 3 3 1	4	8
      4	1 4 6 4 1	5	16
      5	1 5 10 10 5 1	6	32
      6	1 6 15 20 15 6 1	7	64
      7	1 7 21 35 35 21 7 1	8	128
      8	1 8 28 56 70 56 28 8 1	9	256
      9	1 9 36 84 126 126 84 36 9 1	10	512
      10	1 10 45 120 210 252 210 120 45 10 1	11	1024

      
      
   To build the pyramid, start with 1s on the outside. For the inner numbers of the pyramid, add the above two numbers, that is:

   

   Thus, this technique works well with small N's.
   

2. Testing Hypotheses
   Back to our question. Is the coin funny? We toss the coin 10 times and obtain 9 heads.
   1. State the Hypotheses
      Let:
      • p = the probability of a head, and
      • q = the probability of a tail.

      Thus, we can make two mutually exclusive statements or hypotheses as follows. Note, mutually exclusive means that only one of the statements can be true.
      
      

      Hypothesis Name	Meaning	In Symbols	Comments
      Null or HO	The coin is fair	p=q	Always an exact statement
      Alternative or HA or H1	The coin is funny	p¹q	Never an exact statement

      
      
   2. Choose a Significance Level
      The alpha level (a) is the arbitrary level of significance that statisticians have chosen to distinguish probable from improbable. The alpha level chosen in Psychology is typically .05, with .01 or even .001 used in some circumstances.

      Improbable Due to Chance		Probable Due to Chance

   3. Assume the Null Hypothesis
      We do this because the null is an exact statement and therefore testable. In other words, it allows us to compute the relevant probabilities. In the case of the coin, we assume it is fair and compute the probability that the outcome we observed (9 heads in 10 tosses) is due to chance. If the probability of this event occurring due to chance is small (i.e., less than or equal to the alpha level), we will reject the null hypothesis and assert the alternative (that the coin is funny).
      
      
   4. Compute the Probability of the Observed Outcome [Minitab]
      What is the probability of obtaining an event as rare as what we observed? What follows is the binomial distribution for N=10.
      
      

      Possible outcomes	H10T0	H9T1	H8T2	H7T3	H6T4	H5T5	H4T6	H3T7	H2T8	H1T9	H0T10
      P	1/1024	10	45	120	210	252	210	120	45	10	1
      	.001	.010	.044	.117	.205	.246	.205	.117	.044	.010	.001

      The relevant probabilities are shown in color above and are summed below.
      
      

      Event	Probability
      9 heads	.010
      10 heads	.001
      9 tails	.010
      10 tails	.001
      Total	=.022

      
      
   5. Make a Decision
      Compare the probability of the observed outcome to the alpha level and make a decision. Since .022 is less than the alpha level of .05, we conclude that the outcome we have observed is improbable due to chance. In other words, the coin is funny.
      
      
3. Other Relevant Issues.
   
   
1. Directional vs. Non-directional Hypotheses
   
   

   Alternative Hypothesis	Type	Meaning	Test to be used
   p¹q	Non-directional	Coin is biased.	Two-tailed
   p<q	Directional	Coin is biased for T.	One-tailed
   p>q	Directional	Coin is biases for H.	One-tailed

   Nondirectional tests are most common. Directional hypotheses are sometimes used when we have a theory and/or prior data that leads us to such a specific prediction. To keep things simple, in this class, we will use two-tailed tests exclusively.
   
   

2. Accepting, Asserting, and Rejecting
   We never "accept" the null or alternative hypotheses. We either:
   1. Reject the null and "assert" the alternative
      We don’t accept the alternative because we didn’t test it. We tested the null by assuming its truth.
   2. "Fail to reject" the null.
      We don’t accept the null because maybe our test wasn’t sensitive enough to detect a bias in the coin.
      
      
3. Probabilistic Nature of Science
   Given an alpha level of .05, we can expect to reject the null 1 in 20 times when the coin is actually fair. In other words, a fair coin could give 10 heads in 10 tosses, it is just not very probable. Fortunately though, science progresses when the study is replicated (and extended) by the same and other investigators. The probability of making two mistakes would be .05 x .05 = .0025. We will talk more about these "mistakes" in the next chapter. An important moral of this story is that we never used the word "proved" when talking about statistical results.