Social Statistics: Inferential Statistics

The “4 Steps” of Hypothesis Testing:1. State the hypothesis2. Set decision criteria3. Collect data and compute sample

statistic4. Make a decision (accept/reject)

Review

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Types of variables Steps of hypothesis testing Type I and Type II errors

This week

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Between-subjects variable: a characteristic that varies between subjects (e.g. age, gender etc.).

Within-subjects variable: a characteristic of individuals that varies with time. (the same individuals are compared at different points in time).

Types of Variables (Between-subjects vs. Within-subjects)

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The independent variable is the one that is manipulated.

The dependent variable is the one that is observed.

Types of Variables (Independent vs. Dependent )

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A sport psychologist is interested in the relationship between reach and knockouts in boxers. She measures the reach of 20 pro boxers, and then observes the number of knockouts each scores over one year of fighting.

Between-subjects or within-subjects?

Example I

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A researcher is testing the effect of alcohol on memory performance. He gives one group of subjects a bottle of vodka, and another a nonalcoholic substance that tastes like vodka. Each group then learns a list of words, and attempts to recall them. Number of words correctly recalled for each group is recorded.

Between-subjects or within-subjects? What are the independent and dependent variables?

Example II

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A social psychologist is interested in gender differences in math performance. She randomly selects men and women from IU and has them solve a series of equations. Number of equations correctly solved for each participant is recorded.

Between-subjects or within-subjects? What are the independent and dependent

variables?

Example III

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In a survey, one of the questions asks respondents whether they had breakfast today, another asks them whether they are satisfied with their work today.

You found that 80% of the respondents who answered yes on the breakfast question said they were satisfied with their work.

You concluded that eating breakfast will make people feel satisfied about their work.

Justified or not?

A fallacy

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Alternative I (between-subjects) Two groups (same number of people, similar

backgrounds, randomly chosen) One have breakfast; one without having breakfast Test their satisfaction towards work

Alternative II (within-subjects) One group Having breakfast the first day; without having breakfast

the second day Test their satisfaction towards work

Still, there are confounding factors (weather, order, etc.)

A fallacy

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Hypothesis is stated in terms of a population parameter (e.g. μ): assume null unless sufficient evidence to reject it Null Hypothesis: no change in

population parameter, treatment has no effect. H0

Alternate (Scientific/Research) Hypothesis: opposite of H0, treatment has an effect. H1 (directional or non directional)

Step 1: State Hypothesis

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By selecting a probability value (level of significance or alpha level), e.g. α = 0.05, 0.01, 0.001

Critical region is region of extreme sample values (unlikely to be obtained if H0 is true). If sample data fall within the critical region, H0 is rejected.

Determine exact values for boundaries of critical region by using α and unit normal table.

Step 2: Set Criteria

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Critical regions for different values of α

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Select random sample and perform “experiment”.

Compute sample statistic, e.g. sample mean.

Locate sample statistic within hypothesized distribution (use z-score).

Is sample statistic located within the critical region?

Step 3: Collect Data/Statistics

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Possibility: sample statistic is within critical region. Reject H0.

Possibility: sample statistic is not within critical region. Do not reject H0.

Step 4: Decision

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Step 2: Set Criteria

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It is known that corn in Bloomington grows to an average height of μ=72 σ=6 six months after being planted.

We are studying the effect of “Plant Food 6000” on corn growth. We randomly select a sample of 40 seeds from the above population and plant them, using PF-6000 each week for six months. At the end of the six month period, our sample has a height of M=78 inches. Go through the steps of hypothesis testing and draw a conclusion about PF-6000

1. State hypotheses; 2. Chance model/critical region;

3. Collect data; 4. Decision and conclusion

Hypothesis Testing: An Example

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In words Null: PF6000 will not have an effect on

corn growth Alt: PF6000 will have an effect on corn

growth In “code” symbols:

H0 :μ = 72 H1 :μ ≠ 72

Step 1: State Hypotheses

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a) Distribution of Sample Means:

b) Set alpha level α=.05 ∴ zcrit = ±1.96

Step 2: Chance Model and Critical Values

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c) Compute critical values to correspond to zcrit

Step 2: Chance Model and Critical Values

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For the question, this part has already been done for us, we just need to compare this obtained sample mean to our chance model to determine if any discrepancy between our sample and the original population is due to: 1. Sampling Error 2. A true effect of our manipulation

Step 3: Do Experiment

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Mcrit is 70.14 (lower) or 73.86 (upper) If Mobt exceeds either of these critical values

(i.e., is out of the “chance” range, we reject H0. Otherwise, cannot reject H0)

Mobt = 78 Mcrit = 73.86 Mobt exceeds Mcrit ∴ Reject H0

Conclusion: We must reject the null hypothesis that the chemical does not produce a difference. Conclude that PF6000 has an effect on corn growth.

Step 4: Decision and Conclusion

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Accept the null hypothesis

Reject the null hypothesis

The null hypothesis is really true

, you accepted a null when it is true[there is really no difference between the groups]

Type I error: reject a null hypothesis when it is true (represented by the Greek letter alpha, α)[there is really no difference between the groups]

The null hypothesis is really false

Type II error: accepted a false null hypothesis (represented by Greek letter beta, β)[there really are differences between the two groups]

, you rejected the null hypothesis which is false.[there really are differences between the two groups]

Uncertainty and Errors

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Type I and Type II errors are inversely related. Thus by reducing alpha and lowering the risk of making a Type I error, we are increasing the risk of making a Type II error.