Inferential Statistics

• Inferential statistics allow us to infer the characteristic(s) of a population from sample data

• Slightly different terms and symbols are used:

Sample Population“statistics” “parameters” X S

Which Sample is Correct?

Population

RandomSample

= X1

We begin by taking a random sample from the population.

Based on our statistic (e.g., the mean), we would like to make an inference about the nature of the population

= X2

Unfortunately, a different random sample would likely give a different result because of random sampling variability

?

Random Sampling Distributions

• Although we cannot know which sample is the correct one, it would be useful to know how likely a sample value would occur by chance

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Population

RandomSample = X

Random Sampling Distribution

…a relative frequency distribution of a sample statistic, obtained from an unlimited series of sampling experiments, each consisting of a sample of size “n,” randomly selected from the population.

DO NOT FORGET THIS!

An Example

• The random sampling distribution of the mean -- the relative frequency distribution of means, obtained from an unlimited series of sampling experiments, each consisting of a sample of size “n,” randomly selected from the population

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Characteristics of the Random Sampling Distribution of the Mean

x =

Central Limit Theorem holds -- the random sampling distribution of the mean approaches a normal shape as the sample size increases, regardless of the shape of the population distribution

x = √ n

“Standard Error of the Mean”

Hypothesis Testing

How does one go about testing whether a hypothesis is true or not?

For example, suppose the Director of Admissions believes this year’s Freshmen class is above average on the SATs compared to the rest of the country. She calculates the mean SAT for our freshmen and determines it is 981, while the national mean is 960.

Does the data support her hypothesis regarding the Freshmen class?

The Logic of Hypothesis Testing

• Modus tollens - deductive argument of the form, “If P, then Q. Given -Q (not q), you conclude -P (not p)”

– For example,

• If it’s raining, then the streets are wet (if P then Q)

• The streets are not wet (-Q)

• Therefore it’s not raining (-P)

– In the context of testing an hypothesis

• If the hypothesis is true, we expect a certain result

• If the sample result is not what is expected

• The hypothesis is false

• Hypotheses are tested indirectly:

– Two mutually exclusive and exhaustive hypotheses are established

• null hypothesis (H0) - “fake” hypothesis which is assumed to be true, tested directly, and either rejected or not rejected H0: = 960

• alternative hypothesis (H1)- the hypothesis believed to be true by the researcher H1: > 960

• test statistic and sampling distribution are identified

• level of significance (- probability that specifies how rare a sample result must be to reject H0 as being true (e.g., .05 or .01)

• establish region(s) of rejection - area under the curve where H0 is rejected if the sample result should fall within the region (i.e., identifies sample results that are to be considered as -Q)

Criticalvalue

X, reject H0X, do not reject H0

Regionof rejection

• identify critical value(s) - establish the beginning of a region(s) of rejection

The z - test

We must be able to locate the position of our sample result in the random sampling distribution of the mean in order to tell if our sample result is rare enough to consider it as an example of -Q.

That is, we need to calculate an appropriate standard score:

Standard Error of the Mean

X = √ n

z = X - o

X

Critical value = 1.645

1.82, reject H0

Region ofrejection: = .05

The national mean SAT = 960 with = 100. If she used a sample of n = 75, then our z-test statistics is

Standard Error of the Mean

X = √ 75

= 11.55 z = 981 - = 1.82

Assume the Director chooses = .05; the critical value = 1.645.

Rejecting or Not Rejecting Ho

When the calculated value of the test statistic (e.g., z in our example) equals or exceeds the critical value, our decision is to reject Ho.

If, however, the calculated value of the test statistic is less than the critical value, we do not “accept” Ho. Rather, we “do not reject” Ho. That is, there is insufficient evidence to reject Ho.

That may sound like the same thing, but it is not. To “accept” Ho would be to make the fallacious argument called Affirming the Consequent:

If P, then QQ . P

What Does “Significance” Mean?

The assertion of “significance” or “non-significance” is only meaningful in connection with the level used in the test.

In our example, the obtained value of z = 1.82 exceeded the critical value of 1.645 for = .05 and, therefore, Ho was rejected.

If we had decided to use = .01, however, the critical value = 2.33 and we would not have been able to reject Ho.

Statistical vs. Practical Significance

There are two types of significance that must be considered: statistical and practical.

Statistical significance - indicates the Ho was rejected at a specified level.

Practical significance - refers to the real-life importance of the result. That is, does the observed difference between the Ho and H1

really mean anything in practical terms?

More About the Alternative Hypothesis

In some instances our alternative hypothesis will specify one value will be “greater than” or “less than” some other value. In other instances we may believe one value will be “different than” another value, but are unable to predict if it will greater or less than the other value.

Directionalalternative hypothesis: H1: > 960 or H1: < 960

Non-directionalalternative hypothesis: H1: ≠ 960

Estimating the Standard Error of the Mean

There is another problem researchers often encounter: we usually do not know , so we must estimate it from our sample.

Unbiased Estimate of

X2 -(X)2

n = √ n - 1

^

Substituting for , the Estimated Standard Errorof the Mean is given as:

X =

√ n ^ ^

^

The estimate we use is called the “unbiased” estimate of the standard deviation:

The t - test

When we have to estimate , we cannot use the z-test statistic. Instead, we conduct a t-test:

t = X - o

X^

The procedures used to test the hypothesis are the same except we also need to use a different random sampling distribution: Student’s t Distribution.

Student’s t Distribution

Student’s t distribution actually consists of many distributions, each differing in the number of degrees of freedom - the number of scores free to vary when some constraint has been placed on the data.

That is, since x = 0, then n-1 observations are free to vary, since the last observation is fixed.

For example, if you have the following deviation scores:-1, -1, +1, and ? = 0,

then ? must equal +1.

Student’s t Distribution

When we have an infinitely large sample, the t distribution is the same as the z distribution. However, as the sample size decreases (as well as the number of degrees of freedom), the distribution has greater variability.

As a result, we must go further away from the mean to find critical values.

df =

df = 20

df = 4

Level of Significance vs. p-value

There is often some confusion between the level of significance () and the p-value.

Level of significance - specifies how rare a sample result must be in order to reject Ho. It is an independent criterion for evaluating a sample result prior to conducting the analysis.

p-value - how rare the sample result would be if Ho were true. It is the probability, if Ho were true, of observing a sample result as deviant or more deviant than the result obtained.

Correct acceptance1 -

Type II error

Type I error

Correct rejection1-

True SituationHo true Ho false

Do not reject Ho

Researcher Decision

Reject Ho

Type I and Type II Errors

The decision to reject or not reject Ho is done on the basis of probabilities. Therefore, your decision make be incorrect.

= .05

Type I and Type II Errors (con’t)

The figure below depicts the distributions for Ho and H1 to illustrate the relationships between and .

o = 960 = 975

1 - 1 -

The Basic Experiment

Population

RandomSampling

Ideally, we would like to have a sample to study which represents the entire population. That can be accomplished if we use “random sampling” to select our sample.

RandomSample

Convenience

Sample

Random Assignment

Unfortunately, we are seldom able to obtain such a sample. We must, therefore, often use subjects who are readily available -- a “convenience sample” -- and split them into two groups using a technique called “random assignment.”

Control GroupExperimental

Group

Control GroupExperimental

Group

PresentIndependent Variable

MeasureDependent Variable

CompareGroups

An Example

ConvenienceSample:

Students in Gen. Psych. class

Control GroupExperimental

Group

Random Assignment

Have you ever wondered whether those “Highlighters” help you study? Let’s see how we could develop an experiment to test the following hypothesis:

Highlighters facilitate memory of facts read from textbooks.

All subjects will be given several pages to read. After they have done so, they will bedismissed and asked to return to the experimental lab the next day.

Control Group Experimental Group

Present Independent VariableAvailability of Highlighter

MeasureDependent VariableNumber of correctlyrecalled facts on quiz

CompareGroups

No Highlighter Highlighter Available

Each subject is given five pages from an Intro Psych text and told to read thepages carefully because they will be tested on the material. The subjects aredismissed after they finish reading and asked to return the next day.