Psychology 320: Psychological Statistics

Professor: Howard B. Lee

Lecture Notes

Week 6 : Chapter 8

Lecture 15

Inferential Statistics

Hypothesis Testing:

Population: The entire set of possible observations that may be made on the statistical universe. Although the term "universe" typically refers to the elementary units themselves and the term "population" refers to the observations made on the units, the two terms are frequently used interchangeably. In this book, the term population is used to refer to both the elementary units and the observation. The population is defined by the research question.

Sample: A collection of numbers usually drawn at random from a population of numbers which represent a portion of the statistical population. Generally, a sample consists of fewer elementary units or observations than are contained in the population. The sample is a subset of the population.

Population ---> "All" ---> all entities covered in a research question.

What is defined as the population in one study can serve as a sample in another.

Ex. Study 1:
population ---> all CSUN students
sample -------> students in Psych. 320
popsamp1

Study 2:
population ---> all CSU students (all campuses in the CSU system )
sample--------> students at CSUN

The two studies illustrated above involve finite and countable data. They illustrate that the population is determined by what you define it as in a given experiment.

Parameter: A number derived from knowledge of the entire population, such as the minimum of the population, or the mean of the population. Parameters are characteristics in a population.

Statistic: A number derived from observing a sample from a population of numbers. A value computed from a sample.

Greek letters are usually used to represent population parameters.

Example: Greek symbols
When talking about inferential statistics, never use the word "PROVE". Substitute prove with "SHOW" or "DEMONSTRATE".

Hypothesis testing (Inferential Statistics ):

Economics causes a problem when you want to measure the entire population. It costs too much!
inference

SCR = Silicon Controlled Rectifier

Color Organ:
Motorola SCRs were tested individually. Texas Instruments (TI) SCRs were "lot tested". TI took a sample from the population, tested these, and then made an inference about the population.

Political Polls:
Gallup
Nielsen
inference3

Gallup and Nielsen polls let you create the inference. The arrow is very faint. They don't make an inference.

The psychological behavior of these polls is that people would rather be on the winning team than the loosing one. Awareness of the trends pointed out in the polls allows people to make their own inferences about who is the winner or will be. Poor sampling means that the sample is not representative of the population. A poor sample leads to a poor inference.

John Watson = Father of Behaviorism
Albert and the Rabbit
He went to work in advertising industry as many psychologists did.

Hypothesis: A statement about some population parameter that is to be tested for its correctness.

Alternative Hypothesis: The negative of the null hypothesis, also known as the research hypothesis.

Type 1 Error: Rejecting the null hypothesis when it is correct.

Type 2 Error: Accepting the null hypothesis when it is false.

Critical Value: The value of a test statistic that divides the acceptance region from the critical region.

Test Statistic: The statistic used to test a hypothesis about a population parameter.

Rejection Region: Same as the critical region. The critical region is that portion of the area under the sampling distribution of a test statistic that corresponds to rejection of the null hypothesis.

Alpha: The probability of making a Type 1 error. The significance level of a test.

Beta: The probability of making a Type 2 error.

Lecture 16

4 Steps to Hypothesis Testing:

Statement of the null hypothesis.
Statement of alternative hypothesis.
Test Statistic ( requires computations ).
Decision Rule ( requires the use of tables ).
Conclusion - relates the decision back to the original statement of the problem.

The alternative hypothesis, a.k.a. the research hypothesis, is the hypothesis (the actual or real one) of interest. It cannot be tested directly. Because of this, you must develop a null hypothesis which is the opposite of the alternative hypothesis. The null hypothesis can be tested.

The goal is to gather enough data or information to show that the null hypothesis is not true and, as a result, leads one to believe that the alternative is true.

Ex. Criminal Justice System:

The prosecutor claims that "the person is not guilty" (the null hypothesis). This is what you assume is true.
"The person is guilty" is the alternative hypothesis.
The prosecutor must build up enough evidence against the person to show that the null hypothesis is not true. This leads the jury to believe that the alternative hypothesis is true. This evidence/ or data comes in the form of the test statistic.

Ex. A coin is tossed 100 times.

Observed data Expected data

52 50

48 50

Observed data	Expected data
52	50
48	50

The null hypothesis: The coin is fair.
The alternative hypothesis: The coin is not fair.

Error:
Type 2 error: You said that the coin is fair when in fact the coin is unfair (accepted the null H when the null H is false).
Type 1 error: You said that the coin is unfair when in reality the coin is fair (you rejected the null H when the null H is true).

There is a preoccupation with Type 1 error in psychological research. You would want to minimize this type of error.

Probability:
Beta = probability of making a Type 2 error.
Alpha = probability of making a Type 1 error.

Lecture 17

Decision Reality

appendicitis no appendicitis

operate correct Type 2 error

not operate Type 1 error correct

Decision		Reality
appendicitis	no appendicitis
operate	correct	Type 2 error
not operate	Type 1 error	correct

	Reality
	fair	not fair
fair	correct	Type 2 error
not fair	Type 1 error	correct

	Reality
	null H: true	null H: false
accept null	correct	Type 2 error
not accept null	Type 1 error	correct

It is appropriate to say "reject the null hypothesis" when you have proper evidence but because no information is given about Beta, it is inappropriate to say "accept the null hypothesis".

In the criminal justice system, "innocent" is not equivalent to "not guilty". When you are found not guilty this means that they did not have sufficient evidence to convict you.

Theoretically, it is not true to accept the null hyp. if you do not have information about Beta. It is like saying the person is innocent when found not guilty.

Assume the null hypothesis to be true, then gather evidence to show that it is not true. This leads to believing the alternative hypothesis is true.

null H: coin is fair. ( has "=" sign )
alternative H: coin is not fair. (does not have "=" sign)

If a coin is fair, then the # of heads = the # of tails. Ph = .5 = Pt
If coin is not fair, then the # of heads is not = the # of tails. Ph and Pt are not = to .5.

How many different values can be stated for Ph not = .5?
An infinite number!

If you wanted to test the alternative H directly, you would have to test it with every possible value to show that the coin is not fair. This is next to impossible to do!

The null H is also known as the " exact hypothesis" and the alternative is known as the "inexact hypothesis".

The Central Limit Theorem: Regardless of the shape of the population distribution, if all samples of size n are drawn from the population, the sample means will be approximately normally distributed (as n gets larger).

Skinnerians do not use inferential statistics. They use learning curves and descriptive statistics instead.

Ex. Population = (1, 2, 3 , 4 ) mean = 2.5
This is not normally distributed (it is uniformly distributed ).

unidist

Take from the population all samples of size n = 2.
( 1, 2 ) mean = 1.5 ( 2, 3 ) mean = 2.5
( 1, 3 ) mean = 2.0 ( 2, 4 ) mean = 3
( 1, 4 ) mean = 2.5 ( 3, 4 ) mean = 3.5
Plot the means:

sampdist

Sampling distribution, the mean of the sampling means is equal to the population mean.