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Gallagher Theater

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Row C13 12 11 1015 14 6 58 79

Row D13 12 11 1015 1416 6 58 7920 19 18 17Row D

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Row L13 12 11 1015 1416 6 58 7920 19 18 17Row L

Row M13 12 11 1015 14 6 58 7919 18 17 16Row M

Row N13 12 11 1015 1416 6 58 7920 19 18 17Row N

Row O13 12 11 1015 14 6 58 7919 18 17 16Row O

Row P13 12 11 1015 1416 6 58 7920 19 18 17Row P

Row Q13 12 11 10 6 58 7916 15 14Row Q 4

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26Left-Handed DesksA14, B16, B20, C19, D16, D20, E15, E19, F16, F20, G19, H16, H20, I15, J16, J20, K19, L16, L20, M15, M19,

N16, P20, Q13, Q16, S4

5 Broken DesksB9, E12,

G9, H3, M17

Need LabelsB5, E1, I16, J17, K8, M4, O1, P16

Left handed

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table

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Social Sciences 100

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27

MGMT 276: Statistical Inference in Management

Fall, 2014

Green sheets

Reminder

Talking or whispering to your neighbor can be a problem for us – please

consider writing short notes.

A note on doodling

Before our next exam (November 6th) Lind (10 – 12)

Chapter 10: One sample Tests of HypothesisChapter 11: Two sample Tests of HypothesisChapter 12: Analysis of Variance

Plous (2, 3, & 4)Chapter 2: Cognitive Dissonance Chapter 3: Memory and Hindsight BiasChapter 4: Context Dependence

Schedule of readings

On class website: Please print and complete homework worksheets

Assignment 14: Hypothesis Testing using t-tests Due: Thursday, October 30th

Assignments 15 & 16: Hypothesis Testing using t-tests Due: Tuesday, November 4th

Homework

By the end of lecture today 10/30/14

Use this as your study guide

Logic of hypothesis testingSteps for hypothesis testing

Levels of significance (Levels of alpha)what does p < 0.05 mean?what does p < 0.01 mean?

Hypothesis testing with t-scores (one-sample)Hypothesis testing with t-scores (two independent samples)

Constructing brief, complete summary statements

..A note on z scores, and t score:

Difference between means

Variabilityof curve(s)

Difference between means

• Numerator is always distance between means (how far away the distributions are or “effect size”)

• Denominator is always measure of variability(how wide or much overlap there is between distributions)

Variability of curve(s)(within group variability)

Revie

w

.A note on variability versus effect size

Difference between means

Variabilityof curve(s)

Variability of curve(s)(within group variability)

Difference between means

Revie

w

..Difference

between means

Variabilityof curve(s)

Variability of curve(s)(within group variability)

Difference between means

A note on variability versus effect size

Revie

w

.

Effect size is considered relativeto variability of distributions

1. Larger variance harder to find significant difference

Treatment

Effect

Treatment

Effect

2. Smaller variance easier to find significant difference

x

x

.

Effect size is considered relativeto variability of distributions

Treatment

Effect

Treatment

Effect

x

x

Variability of curve(s)(within group variability)

Difference between means

Five steps to hypothesis testing

Step 1: Identify the research problem (hypothesis)

Describe the null and alternative hypotheses

Step 2: Decision rule: find “critical z” score

• Alpha level? (α = .05 or .01)?

Step 3: Calculations

Step 4: Make decision whether or not to reject null hypothesisIf observed z (or t) is bigger then critical z (or t) then reject null

Step 5: Conclusion - tie findings back in to research problem

Population versus sample standard deviation

Population versus sample standard deviation

How is a t score different than a z score?

• One versus two-tailed test

Comparing z score distributions with t-score distributions

Similarities include:

Using bell-shaped distributions to make confidence interval estimations and decisionsin hypothesis testing

Use table to find areas under the curve(different table, though – areas often differ from z scores)

z-scores

t-scoresSummary of 2 main differences:• We are now estimating standard deviation from the sample

(We don’t know population standard deviation)• We have to deal with degrees of freedom

.

Interpreting t-table

Technically, we have a different t-distribution for each sample size

This t-table summarizes the most useful values for several distributions

n = 17

n = 5

This t-table presents useful values for

distributions (organized by degrees of freedom)

1.96 2.581.64

Remember these useful values for z-scores?

We use degrees of freedom (df) to

approximate sample size

Each curve is based on its own degrees of

freedom (df) - based on sample size, and its

own table tying together t-scores with area under the curve

Comparison of z and t

• For very small samples, t-values differ substantially from the normal.

• As degrees of freedom increase, the t-values approach the normal z-values.

• For example, for n = 31, the degrees of freedom are:

What would the t-value be for a 90% confidence interval?

n - 1 = 31 – 1 = 30

df

Degrees of Freedom

Degrees of Freedom (d.f.) is a parameter based on the sample size that is used to determine the value of the t statistic.

Degrees of freedom tell how many observations are used to calculate s, less the number of intermediate estimates used in the calculation.

Area between two scores

Area between two scores

Area beyond two scores

(out in tails)

Area beyond two scores

(out in tails)

Area in each tail

(out in tails)

Area in each tail

(out in tails)df

Area between two scores

Area between two scores

Area beyond two scores

(out in tails)

Area beyond two scores

(out in tails)

Area in each tail

(out in tails)

Area in each tail

(out in tails)

Notice with large sample size it is same values as

z-score.

1.96 2.581.64

Remember these useful values for z-

scores?

df

A quick re-visit with the law of large numbers

Relationship between • increased sample size• decreased variability• smaller “critical values”

As n goes upvariability goes

down

Law of large numbers: As the number of measurementsincreases the data becomes more stable and a better

approximation of the true signal (e.g. mean)

As the number of observations (n) increases or the number of times the experiment is performed, the signal will become more clear (static cancels out)

http://www.youtube.com/watch?v=ne6tB2KiZuk

With only a few people any little error is noticed (becomes exaggerated when we look at whole

group)

With many people any little error is corrected (becomes minimized when we look at whole

group)

Crowd sourcing for predicting future events

Wisdom of CrowdsFrancis Galton (1906)

Revisit: Law of large numbers• Deviation scores / Error term

- how far away the individual scores (guesses) are from the true score

• Mean (The over-estimates and under-estimates balance each other out)

http://www.npr.org/blogs/parallels/2014/04/02/297839429/-so-you-think-youre-smarter-than-a-cia-agent

Comparing z score distributions with t-score distributions

Differences include:

1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample

Critical t (just like critical z)

separates common from rare scores

Critical t used to define both common scores “confidence interval”

and rare scores “region of rejection”

Comparing z score distributions with t-score distributions

2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n)

Please notice: as sample sizes get smaller, the tails

get thicker. As sample sizes get bigger tails get

thinner and look more like the z-distribution

Differences include:

1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample

Comparing z score distributions with t-score distributions

2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n)

Differences include:

1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample

Please notice: as sample sizes get smaller, the tails

get thicker. As sample sizes get bigger tails get

thinner and look more like the z-distribution

Comparing z score distributions with t-score distributions

2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n)

Differences include:

1) We use t-distribution when we don’t know standard deviation of population, and have to estimate it from our sample

3) Because the shape changes, the relationship between the scores and proportions under the curve change (So, we would have a different table for all the different possible n’s but just the important ones are summarized in our t-table)

Please note: Once sample sizes get big

enough the t distribution (curve) starts to look

exactly like the z distribution (curve) scores

mean + z σ = 30 ± (1.96)(2)

mean + z σ = 30 ± (2.58)(2)

26.08 < µ < 33.92

24.84 < µ < 35.16

95%

99%

Melvin

Mark

Melvin

Difference not due sample size because both samples same size

Difference not due population variability because same population

Yes! Difference is due to sloppiness and random error in Melvin’s sample

Melvin

Ho: µ = 5Ha: µ ≠ 5

Bags of potatoes from that plant are not different from other plantsBags of potatoes from that plant are different from other plants

Two tailed test(α = .05)1.96

6 – 5.25= 4.0

116√

= .25

4.01.96-1.96

14=

z- score : because we know the population standard deviation

YesYesYes

These three will always

match Probability of Type I error is always equal

to alpha.05

Because theobserved z (4.0 ) is bigger

than critical z (1.96)

1.64No

Because observed z (4.0) is still bigger than critical z (1.64)

2.58

there is a difference

NoBecause observed z (4.0) is still bigger than critical z(2.58)

there is no differencethere is notthere is

1.962.58

Two tailed test(α = .05)

Critical t(15) = 2.13189 - 85

616√

2.667

t- score : because we don’t know the population standard deviation

n – 1 =16 – 1 = 15

2.13-2.13

YesYesYes

These three will always

match Probability of Type I error is always equal

to alpha.05

Because theobserved z (2.67) is bigger

than critical z (2.13)

1.753No

Because observed t (2.67) is still bigger than critical t (1.753)

2.947

consultant did improve morale

YesBecause observed t (2.67) is not bigger than critical t(2.947)

consultant did not improve moraleshe did notshe did

2.1312.947

NoNoNo

These three will always

match

The average weight of bags of potatoes from this particular plantis 6 pounds, while the average weight for population is 5 pounds.A z-test was completed and this difference was found to be statistically significant. We should fix the plant. (z = 4.0; p<0.05)

Start summary with two means (based on DV) for two levels of the IV Describe type of test

(z-test versus t-test) with brief overview of

results

Finish with statistical summary

z = 4.0; p < 0.05

Or if it *were not* significant:

z = 1.2 ; n.s.

Value of observed statistic

n.s. = “not significant”p<0.05 =

“significant”

The average job-satisfaction score was 89 for the employees who wentOn the retreat, while the average score for population is 85. A t-testwas completed and this difference was found to be statistically significant. We should hire the consultant. (t(15) = 2.67; p<0.05)

Start summary with two means (based on DV) for two levels of the IV Describe type of test

(z-test versus t-test) with brief overview of

results

Finish with statistical summary

t(15) = 2.67; p < 0.05

Or if it *were not* significant:

t(15) = 1.07; n.s.

df

Value of observed statistic

n.s. = “not significant”p<0.05 =

“significant”

..A note on z scores, and t score:

Difference between means

Variabilityof curve(s)

Difference between means

Difference between means

• Numerator is always distance between means (how far away the distributions are)

• Denominator is always measure of variability(how wide or much overlap there is between distributions)

Variabilityof curve(s)Variabilityof curve(s)

Five steps to hypothesis testing

Step 1: Identify the research problem (hypothesis)

Describe the null and alternative hypotheses

Step 2: Decision rule

• Alpha level? (α = .05 or .01)?

Step 3: Calculations

Step 4: Make decision whether or not to reject null hypothesisIf observed z (or t) is bigger then critical z (or t) then reject null

Step 5: Conclusion - tie findings back in to research problem

• Critical statistic (e.g. z or t) value?

How is a single sample t-test different than two sample t-test?

Single sample standard deviation versus average

standard deviation for two samples

How is a single sample t-test most similar to the two sample t-test?

Single sample has one “n” while two samples will have

an “n” for each sample

Independent samples t-test

Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha = .05

Big Meal222525

Small meal192321

Mean= 24

Mean= 21

t =x1 – x2

variabilityt =

24 – 21variability

Got to figure this part out: We want to average from 2 samples - Call it

“pooled”

Are the two means significantly different from each other, or is

the difference just due to chance?

Hypothesis testing

Step 1: Identify the research problem

Step 2: Describe the null and alternative hypotheses

Did the size of the meal affect the learning / test scores?

Step 3: Decision ruleα = .05 Two tailed test

Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1)= (3 - 1) + (3 – 1) = 4

n1 = 3; n2 = 3

Critical t(4) = 2.776

Step 4: Calculate observed t score

Notice: Two different

ways to think about it

α= .05

(df) = 4

Critical t(4)

= 2.776

two tail test

3

4

Mean= 24

SquaredDeviation

440

Σ = 8

Big Meal222525

Small meal192321

Big MealDeviation

From mean-211

Squareddeviation

411

Mean= 21Small MealDeviation

From mean-2 20

Σ = 6

= 3.5

S2pooled =

(n1 – 1) s12 + (n2 – 1) s2

2

n1 + n2 - 2

S2pooled =

(3 – 1) (3) + (3 – 1) (4)

31 + 32 - 2

621

82

1

22

Notice: s2 = 3.0

Notice: s2 = 4.0

Notice: Simple Average = 3.5

Mean= 24 Squared

Deviation440

Σ = 8

Participant123

Big Meal222525

Small meal192321

Big MealDeviation

From mean-211

Squareddeviation

411

Mean= 21 Small Meal

DeviationFrom mean

-220

Σ = 6

=24 – 21

1.5275= 1.964

S2p = 3.5

24 - 21

3.5 3.5

3 3

Observed t

1.964 is not larger than 2.776 so, we do not reject the null hypothesist(4) = 1.964; n.s.

Observed t = 1.964

Critical t = 2.776

Conclusion: There appears to be no difference between the groups

How to report the findingsfor a t-test

One paragraph summary of this study. Describe the IV & DV.Present the two means, which type of test was conducted, and the statistical results.

Observed t = 1.964df = 4

Mean of big meal was

24

Mean of small meal

was 21

We compared test scores for large and small meals.The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two typesof meals t(4) = 1.964; n.s.

Start summary with two means (based on DV) for two levels of the IV

Describe type of test (t-test versus anova) with brief overview of

results

Finish with statistical summary

t(4) = 1.96; ns

Or if it *were* significant:

t(9) = 3.93; p < 0.05

Type of test with

degrees of freedom

Value of observed statistic

n.s. = “not significant”p<0.05 =

“significant”

n.s. = “not significant”p<0.05 =

“significant”

We compared test scores for large and small meals. The mean test scores for the big meal was 24, and was 21 for the small meal. A t-test was calculated and there appears to be no significant difference in test scores between the two types of meals, t(4) = 1.964; n.s.

Start summary with two means (based on DV) for two levels of the IV

Describe type of test (t-test versus anova) with brief overview of

results

Finish with statistical summary

t(4) = 1.96; ns

Or if it *were* significant:

t(9) = 3.93; p < 0.05

Type of test with

degrees of freedom

Value of observed statistic

n.s. = “not significant”p<0.05 =

“significant”