Tuesday, April 8

Inferential statistics – Part 2Inferential statistics – Part 2 Hypothesis testingHypothesis testing Statistical significanceStatistical significance continued….continued….

What you want to know is What you want to know is

what is going on in the population?what is going on in the population? All you have is sample dataAll you have is sample data

Your research hypothesis states there is a Your research hypothesis states there is a difference between groups difference between groups

Null hypothesis states there is NO difference Null hypothesis states there is NO difference between groupsbetween groups

Even though your sample data show some Even though your sample data show some difference between groups, there is a chance difference between groups, there is a chance that there is no difference in populationthat there is no difference in population

Sample Population

?Male GPA= 3.3

Female GPA = 3.6 Inferential

N=100

Parameter

Ha: Female students have higher GPA than Male students at UH

Three Possibilities Females really have higher Females really have higher

GPA than MalesGPA than Males

Females with higher GPA Females with higher GPA are disproportionately are disproportionately selected because of selected because of sampling biassampling bias

Females’ GPA happened to Females’ GPA happened to be higher in this particular be higher in this particular sample due to sample due to random random sampling errorsampling error

Male GPA= 3.3

Female GPA =3.6

Logic of Hypothesis Testing

Statistical tests used in hypothesis testing deal with the probability of a particular event occurring by chance.

Is the result common or a rare occurrence

if only chance is operating?

A score (or result of a statistical test) is “Significant”

if score is unlikely to occur on basis of chance alone.

The “Level of Significance” is a cutoff point for determining significantly rare or unusual scores.

Scores outside the middle 95% of a distribution are considered “Rare” when we adopt the standard

“5% Level of Significance”

This level of significance can be written as:

p = .05

Level of Significance

P < .05Reject Null Hypothesis (H0)

Support Your Hypothesis (Ha)

Decision Rules

Reject Ho (accept Ha) when

sample statistic is statistically significant at

chosen p level, otherwise accept Ho (reject Ha).

Possible errors:

• You reject Null Hypothesis when in fact it is true,

Type I Error, or Error of Rashness.

B. You accept Null Hypothesis when in fact it is false,

Type II Error, or Error of Caution.

Correct

Correct

Ho (no fire) Ha (fire)

Ho = null hypothesis = there is NO fire

Ha = alternative hyp. = there IS a FIRE

Accept Ho

(no fire)

Type IIerror

Type I errorReject Ho

(alarm)

True State

What Statistics CANNOT do Statistics CANNOT THINK or reason.Statistics CANNOT THINK or reason. It’s It’s

only only youyou who can think.who can think. Something can be statistically significant, yet Something can be statistically significant, yet

be meaningless. be meaningless.

Statistics - about probability, thus canStatistics - about probability, thus canNOT NOT prove prove your argument. Only your argument. Only supportsupport it. it.

Reject null hypothesis if probability is Reject null hypothesis if probability is <.05<.05 (probability of Type I error < than .05)(probability of Type I error < than .05)

Statistics Statistics can NOT show causalitycan NOT show causality; can show ; can show co-occurrence, which only co-occurrence, which only impliesimplies causality. causality.

When to use various statistics

ParametricParametric Interval or ratio dataInterval or ratio data

Non-parametricNon-parametric Use with non-Use with non-

interval/ratio data (i.e., interval/ratio data (i.e., ordinal and nominal)ordinal and nominal)

Parametric Tests

Used with data w/ mean score or standard Used with data w/ mean score or standard deviation.deviation.

t-test, ANOVA and Pearson’s Correlation r.t-test, ANOVA and Pearson’s Correlation r.

Use a Use a t-testt-test to compare mean differences to compare mean differences between two groups (e.g., male/female and between two groups (e.g., male/female and married/single).married/single).

Parametric Tests

use use ANalysis Of VArianceANalysis Of VAriance (ANOVA) to (ANOVA) to compare more than two groups (such as compare more than two groups (such as age and family income) to get probability age and family income) to get probability scores for the overall group differences. scores for the overall group differences.

Use a Use a Post Hoc TestsPost Hoc Tests to identify which to identify which subgroups differ significantly from each subgroups differ significantly from each other.other.

When comparing two groups on MEAN SCORES use the t-test.

t =

+1 2

M ean - M ean

2SD

n

2SD

n 2

2

1

1

When comparing more than two groups on MEAN SCORES,

use Analysis of Variance (ANOVA)

The computer will do all the work!

T-test

If If pp<.05<.05, we conclude that two groups are , we conclude that two groups are drawn from populations with different drawn from populations with different distribution (distribution (reject Hreject H00) at 95% confidence ) at 95% confidence levellevel

Product RecognitionBest Recognized Shapes

VAS-J SCOPE J&J-P OLAY J&J-S JERGNS TUMS VAS-L SECRT MENN ARID0

20

40

60

80

100Percent

Product RecognitionLeast Recognized Shapes

SEL-B CREST M.O.M. CHLOR. SKIN-B VITAL BARBI RT.GRD. KAO DESE S&S-L0

20

40

60

80

100Percent

Product RecognitionComparison of Total Scores

Male Female 17-26 27-33 34+ Single Married 0-15 16-35 36+9

10

11

12

13

14Mean Score

Gender Age Marital Status Family Incomep=.02 p=.001 p=.91 p=.004

Is there a meaningful difference between subgroups?Question:

Answer:Use Inferential Statistics to help you decide if differences could be due to chance, or are they likely a true difference between groups.

If differences are too large to be due to chance, then there is a Significant Difference between the groups.

We know the probability that our conclusions may be incorrect.

Males:

Mean = 11.3

SD = 2.8

n = 135

Females:

Mean = 12.6

SD = 3.4

n = 165

11.3 12.6

12

Mean scores reflect real difference

between genders.

Mean scores are just chance differences from

a single distribution.

** Accept Ha

Accept Ho

p = .02

Married:

Mean = 11.9

SD = 3.8

n = 96

Single:

Mean = 12.1

SD = 4.3

n = 204

Mean scores reflect real difference

between groups.

Mean scores are just chance differences

from a single distribution.

Accept Ha

**Accept Ho

p = .9111.9 12.1

12

Decision Making2 ways you can be “right”

Your inference based onsample data:

Reject H0

Accept H0

RealityH0 true H0 false

Decision Making2 ways you can be wrong

XType IIerror

XType IerrorYour inference

based onsample data:

Reject H0

Accept H0

RealityH0 true H0 false

Rating the importance of the reason for purchasing clothing "You need that piece

of clothing" in a 7-point scale

4.92

5.52

1

2

3

4

5

6

7

Female

Male

Most Important

Neutral

Least Important

p=0.038

Correlation - Measures of Association

Pearson’s Correlation CoefficientPearson’s Correlation Coefficient

both variables interval / ratio databoth variables interval / ratio data

one variable interval and other nominalone variable interval and other nominal

Non-parametric:Non-parametric: Spearman’s Rank Order CorrelationSpearman’s Rank Order Correlation

with ordinal datawith ordinal data Phi coefficientPhi coefficient

Both variables are dichotomous (2 choices Both variables are dichotomous (2 choices only)only)

Measures of Association Measures of associationMeasures of association examines the examines the

relationship between two variables = relationship between two variables = bivariate analysisbivariate analysis

Measures of association involves Measures of association involves the the significance testsignificance test

Significance test examines Significance test examines the probability the probability of TYPE I errorof TYPE I error

Conventionally we reject the null Conventionally we reject the null hypothesis if probability is hypothesis if probability is <.05<.05 (probability (probability of TYPE I error is smaller than .05)of TYPE I error is smaller than .05)

Measures of Association

Ordinal Intervl/Ratio

Nominal

Ordinal

Intervl/Ratio

Correlation (Spearman)

Correlation (Spearman)

Correlation (Pearson)

Measures of association to use depends on which level your variables are measured

Nominal

Phi coefficient

HEIGHT SMART

Men Women Men Women

Very Unimportant Very Unimportant

Unimportant Unimportant

Neutral Neutral

Important Important

Very Important Very Important

Total Total

TEETH

Men Women Men Women

Very Unimportant Very Unimportant

Unimportant Unimportant

Neutral Neutral

Important Important

Very Important Very Important

Total Total

When selecting a potential dating partner, how important are the following characteristics?

HEIGHT F>M p=.001

Men Women

Very Unimportant 0 2 1

Unimportant 22 2 12

Neutral 23 12 18

Important 47 58 53

Very Important 6 26 16

Total 100 100

Importance of HEIGHTBy Gender

p = .001

Very Unimportant Unimportant Neutral Important Very Important0

10

20

30

40

50

60Percent

Men Women

SMART F>M p=.05

Men Women

Very Unimportant 0 0 0

Unimportant 0 0 0

Neutral 10 6 8

Important 58 38 48

Very Important 32 56 44

Total 100 100

Importance of SMARTBy Gender

p = .05

Very Unimportant Unimportant Neutral Important Very Important0

10

20

30

40

50

60Percent

Men Women

TEETH F=M p=.88

Men Women

Very Unimportant 0 0 0

Unimportant 8 12 10

Neutral 26 28 27

Important 44 38 41

Very Important 22 22 22

Total 100 100

Importance of TEETHBy Gender

p = .88

Very Unimportant Unimportant Neutral Important Very Important0

10

20

30

40

50Percent

Men Women