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2. MGMT 276: Statistical Inference in Management Spring 2015
Welcome

4. Schedule of readings Before our next exam (April 14th) Lind (10 – 12)
Chapter 10: One sample Tests of Hypothesis
Chapter 11: Two sample Tests of Hypothesis
Chapter 12: Analysis of Variance
Plous (2, 3, & 4)
Chapter 2: Cognitive Dissonance
Chapter 3: Memory and Hindsight Bias
Chapter 4: Context Dependence

5. By the end of lecture today 3/26/15
Logic of hypothesis testing
Steps 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

6. Homework due – Tuesday (April 7th)
On class website:
Please print and complete homework worksheet #13 & 14
Hypothesis testing using t-tests
Please note:
Because this homework is longer than most, it is worth two assignments

7. 26.08 < µ < 33.92 mean + z σ = 30 ± (1.96)(2)
95%
< µ < 33.92
mean + z σ = 30 ± (1.96)(2)
99%
< µ < 35.16
mean + z σ = 30 ± (2.58)(2)

8. Melvin
Melvin
Mark
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

9. 6 – 5 = 4.0 .25 Two tailed test 1.96 (α = .05) 1 1 = = .25 16 4 √ 4.0
z- score : because we know the population standard deviation
Ho: µ = 5
Bags of potatoes from that plant are not different from other plants
Ha: µ ≠ 5
Bags of potatoes from that plant are different from other plants
Two tailed test
1.96
(α = .05) 1 1 =
= .25
6 – 5 16 4 √
= 4.0
.25
4.0
-1.96
1.96

10. Because the observed z (4.0 ) is bigger than critical z (1.96)
These three will always match
Yes
Yes
Probability of Type I error is always equal to alpha
Yes
.05
1.64 No
Because observed z (4.0) is still bigger than critical z (1.64)
2.58 No
Because observed z (4.0) is still bigger than critical z(2.58)
there is a difference
there is not
there is no difference
there is
1.96
2.58

11. 89 - 85 Two tailed test (α = .05) n – 1 =16 – 1 = 15
-2.13
2.13
t- score : because we don’t know the population standard deviation
Two tailed test
(α = .05)
n – 1 =16 – 1 = 15
Critical t(15) = 2.131
2.667 6 √ 16

12. two tail test
α= .05
(df) = 15
Critical t(15)
= 2.131 12

13. Because the observed z (2.67) is bigger than critical z (2.13)
These three will always match
Yes
Yes
Probability of Type I error is always equal to alpha
Yes
.05
1.753 No
Because observed t (2.67) is still bigger than critical t (1.753)
2.947
Yes
Because observed t (2.67) is not bigger than critical t(2.947) No
These three will always match No No
consultant did improve morale
she did not
consultant did not improve morale
she did
2.131
2.947

14. Value of observed statistic
Finish with statistical summary z = 4.0; p < 0.05
Or if it *were not* significant:
z = 1.2 ; n.s.
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
n.s. = “not significant”
p<0.05 = “significant”
The average weight of bags of potatoes from this particular plant
is 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)
Value of observed statistic

15. Value of observed statistic
Finish with statistical summary t(15) = 2.67; p < 0.05
Or if it *were not* significant:
t(15) = 1.07; n.s.
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
n.s. = “not significant”
p<0.05 = “significant”
The average job-satisfaction score was 89 for the employees who went
On the retreat, while the average score for population is 85. A t-test
was completed and this difference was found to be statistically
significant. We should hire the consultant. (t(15) = 2.67; p<0.05)
Value of observed statistic df

16. A note on z scores, and t score: . .
A note on z scores, and t score:
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)
Difference between means
Difference between means
Variability of curve(s) (within group variability)
Variability of curve(s)

17. A note on variability versus effect size Difference between means.
A note on variability
versus effect size
Difference between means
Difference between means
Variability of curve(s)
Variability of curve(s) (within group variability)

18. A note on variability versus effect size Difference between means.
A note on variability
versus effect size
Difference between means
Difference between means .
Variability of curve(s)
Variability of curve(s) (within group variability)

19. Effect size is considered relative to variability of distributions.
Effect size is considered relative to variability of distributions
1. Larger variance harder to find significant difference
Treatment
Effect x
Treatment
Effect
2. Smaller variance easier to find significant difference x

20. Effect size is considered relative to variability of distributions.
Effect size is considered relative to variability of distributions
Treatment
Effect x
Difference between means
Treatment
Effect x
Variability of curve(s) (within group variability)

21. Five steps to hypothesis testing
Step 1: Identify the research problem (hypothesis)
How is a t score different than a z score?
Describe the null and alternative hypotheses
Step 2: Decision rule: find “critical z” score
Alpha level? (α = .05 or .01)?
One versus two-tailed test
Step 3: Calculations
Step 4: Make decision whether or not to reject null hypothesis
If observed z (or t) is bigger then critical z (or t) then
reject null
Population versus sample standard deviation
Population versus sample standard deviation
Step 5: Conclusion - tie findings back in to research problem

22. Comparing z score distributions with t-score distributions
z-scores
Similarities include:
Using bell-shaped distributions to make
confidence interval estimations and decisions
in hypothesis testing
Use table to find areas under the curve
(different table, though
– areas often differ from z scores)
t-scores
Summary 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

23. We use degrees of freedom (df) to approximate sample size
Interpreting t-table
We use degrees of freedom (df) to approximate sample size
Technically, we have a different
t-distribution for each sample size
This t-table summarizes the most
useful values for several distributions
This t-table presents useful values for distributions (organized by degrees of freedom)
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
n = 17
n = 5 .
Remember these
useful values for z-scores?
1.64
1.96
2.58

24. 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

25. useful values for z-scores? .
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
Notice with large sample size it is same values as z-score
Remember these
useful values for z-scores? .
1.96
2.58
1.64

26. A quick re-visit with the law of large numbers
Relationship between
increased sample size
decreased variability
smaller “critical values”
As n goes up
variability goes down

27. Law of large numbers: As the number of measurements
increases 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)
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)

28. 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)
1587 pounds
Wisdom of Crowds Francis Galton (1906)
Crowd sourcing for predicting future events

29. Comparing z score distributions with t-score distributions
Differences include:
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”

30. Comparing z score distributions with t-score distributions
Differences include:
We use t-distribution when we don’t know standard deviation
of population, and have to estimate it from our sample
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

31. Comparing z score distributions with t-score distributions
Differences include:
We use t-distribution when we don’t know standard deviation
of population, and have to estimate it from our sample
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

32. Comparing z score distributions with t-score distributions
Please note: Once sample sizes get big enough the t distribution (curve) starts to look exactly like the z distribution (curve) scores
Comparing z score distributions with t-score distributions
Differences include:
We use t-distribution when we don’t know standard deviation
of population, and have to estimate it from our sample
2) The shape of the sampling distribution is very sensitive to small sample sizes (it actually changes shape depending on n)
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)

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

34. 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.

35. A note on z scores, and t score: . .
A note on z scores, and t score:
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)
Difference between means
Difference between means
Difference between means
Variability of curve(s)
Variability of curve(s)
Variability of curve(s)

36. Five steps to hypothesis testing
Step 1: Identify the research problem (hypothesis)
How is a single sample t-test different than two sample t-test?
Describe the null and alternative hypotheses
How is a single sample t-test most
similar to the two sample t-test?
Step 2: Decision rule
Alpha level? (α = .05 or .01)?
Critical statistic (e.g. z or t) value?
Step 3: Calculations
Step 4: Make decision whether or not to reject null hypothesis
If observed z (or t) is bigger then critical z (or t) then
reject null
Single sample standard deviation versus average standard deviation for two samples
Single sample has one “n” while two samples will have an “n” for each sample
Step 5: Conclusion - tie findings back in to research problem

37. Independent samples t-test
Are the two means significantly different from each other, or is the difference just due to chance?
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 Meal 22 25
Small meal 19 23 21
Mean= 21
Mean= 24
Got to figure this part out: We want to average from 2 samples - Call it “pooled”
t =
24 – 21
variability
x1 – x2
t =
variability 37

38. Notice: Two different ways to think about it
Hypothesis testing
Step 1: Identify the research problem
Did the size of the meal affect the learning / test scores?
Step 2: Describe the null and alternative hypotheses
Step 3: Decision rule
α = .05
Two tailed test
n1 = 3; n2 = 3
Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1)
= (3 - 1) + (3 – 1) = 4
Critical t(4)
= 2.776
Step 4: Calculate observed t score
Notice: Two different ways to think about it 38

39. two tail test
α= .05
(df) = 4
Critical t(4)
= 2.776 39

40. Notice: Simple Average = 3.5
Mean= 21
Mean= 24
Big Meal
Deviation
From mean -2 1
Small Meal
Deviation
From mean -2 2
Squared
deviation 4 1
Squared Deviation 4
Big Meal 22 25
Small meal 19 23 21
Σ = 6
Σ = 8 6 3
Notice: s2 = 3.0 1 2 1
Notice: Simple Average = 3.5 8 4
Notice: s2 = 4.0 2 2 2
S2pooled =
(n1 – 1) s12 + (n2 – 1) s22
n1 + n2 - 2
S2pooled =
(3 – 1) (3) + (3 – 1) (4)
= 3.5 40

41. Conclusion: There appears to be no difference between the groups
S2p = 3.5
Mean= 21
Mean= 24
Big Meal
Deviation
From mean -2 1
Small Meal
Deviation
From mean -2 2
Squared
deviation 4 1
Squared Deviation 4
Participant 1 2 3
Big Meal 22 25
Small meal 19 23 21
Σ = 6
Σ = 8 =
24 – 21
1.5275
= 1.964
3.5
3.5 3 3
Observed t =
Observed t
Critical t = 2.776
1.964 is not larger than so, we do not reject the null hypothesis
t(4) = 1.964; n.s.
Conclusion: There appears to be no difference between the groups 41

42. How to report the findings for a t-test
Mean of small meal was 21
How to report the findings for a t-test
Mean of big meal was 24
One paragraph summary of this study. Describe the IV & DV.
Present the two means, which type of test was conducted, and
the statistical results.
Finish with statistical summary t(4) = 1.96; ns
Start summary with two means (based on DV) for two levels of the IV
Observed t =
df = 4
Or if it *were* significant:
t(9) = 3.93; p < 0.05
Describe type of test (t-test versus anova) with brief overview of results
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.
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic 42

43. Type of test with degrees of freedom Value of observed statistic
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.
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic
Start summary with two means (based on DV) for two levels of the IV
Finish with statistical summary t(4) = 1.96; ns
Describe type of test (t-test versus anova) with brief overview of results
Or if it *were* significant:
t(9) = 3.93; p < 0.05 43

44. Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25
Small meal 19 23 21 44

45. Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25
Small meal 19 23 21 45

46. Complete a t-test Mean= 21 Mean= 24 Participant 1 2 3 Big Meal 22 25
Small meal 19 23 21
If checked you’ll want to include the labels in your variable range
If checked, you’ll want to include the labels in your variable range
If checked you’ll want to include the labels in your variable range 46

47. Finding Means
Finding Means 47

48. This is variance for each sample
(Remember, variance is just standard deviation squared)
Please note:
“Pooled variance” is just like the average of the two sample variances, so notice that the average of 3 and 4 is 3.5 48

49. This is “n” for each sample This is “n” for each sample
(Remember, “n” is just number of observations for each sample)
This is “n” for each sample
(Remember, “n” is just number of observations for each sample)
Remember, “degrees of freedom” is just (n-1) for each sample.
So for sample 1: n-1 =3-1 = 2
And for sample 2: n-1=3-1 = 2
Then, df = 2+2=4
df = “degrees of freedom” 49

50. Finding degrees of freedom50

51. Finding Observed t 51

52. Finding Critical t 52

53. Finding Critical t 53

54. Finding p value (Is it less than .05?)54

55. 55

56. Type of test with degrees of freedom Value of observed statistic
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.
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic
Start summary with two means (based on DV) for two levels of the IV
Finish with statistical summary t(4) = 1.96; ns
Describe type of test (t-test versus anova) with brief overview of results
Or if it *were* significant:
t(9) = 3.93; p < 0.05 56

57. Hypothesis testing
α = .05
Step 4: Make decision whether or not to reject null hypothesis
Reject when:
observed stat > critical stat
is not bigger than 2.776
“p” is less than 0.05 (or whatever alpha is)
p = is not less than 0.05
Step 5: Conclusion - tie findings back in to research problem
There was no significant difference, there is no
evidence that size of meal affected test scores 57

58. Type of test with degrees of freedom Value of observed statistic
The mean test score for participants who ate the big meal
was 24, while the mean test score for participants who ate the
small meal was 21. A t-test was completed and there appears
to be no significant difference in the test scores as a function
of the size of the meal, t(4) = 1.96; n.s.
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic
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 58

59. Graphing your t-test results59

60. Graphing your t-test results60

61. Graphing
your t-test results
Chart Layout 61

62. Graphing
your t-test results
Fill out titles 62

63. Independent samples t-test
What if we ran more subjects?
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 Meal 22 25
Small meal 19 23 21
Mean= 21
Mean= 24 63

64. Notice: Additional participants don’t affect this part of the problem
Hypothesis testing
Notice: Additional participants don’t affect this part of the problem
Step 1: Identify the research problem
Did the size of the meal affect the test scores?
Step 2: Describe the null and alternative hypotheses
Ho: The size of the meal has no effect on
test scores
H1: The size of the meal does have an effect on
test scores
One tail or two tail test? 64

65. Notice: Two different ways to think about it
Hypothesis testing
Step 3: Decision rule
α = .05
n1 = 9; n2 = 9
Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1)
= (9 - 1) + (9 – 1) = 16
Degrees of freedom total (df total) = (n total - 2) = 18 – 2 = 16
two tailed test
Notice: Two different ways to think about it
Critical t(16)
= 2.12 65

66. two tail test
α= .05
(df) = 16
Critical t(16)
= 2.12 66

67. 8 8 18 = 1.50 Notice: s2 = 2.25 24 = 1.732 Notice: s2 = 3.0 Mean= 21
Big Meal
Deviation
From mean 2 -1
Small Meal
Deviation
From mean 2 -2
Squared
deviation 4 1
Squared Deviation 4
Big Meal 22 25
Small meal 19 23 21
Σ = 18
Σ = 24
= 1.50 18
Notice: s2 = 2.25 1 8 1
Notice:
Simple Average = 2.625
= 1.732 24
Notice: s2 = 3.0 2 2 8 67

68. Sp = 2.625 S21 = 2.25 S22 = 3.00 S1 = 1.5 S2 = 1.732 Mean= 21 Mean= 24
Big Meal 22 25
Small meal 19 23 21
Sp = 2.625
S21 = 2.25
S22 = 3.00
S1 = 1.5
S2 = 1.732
S2pooled =
(n1 – 1) s12 + (n2 – 1) s22
n1 + n2 - 2
S2pooled =
(9 – 1) (1.50) 2 + (9 – 1) (1.732)2
= 2.625 68

69. Sp = 2.625 S1 = 1.5 S2 = 1.732 Mean= 21 Mean= 24 = 3.928 = 24 – 21
Big Meal 22 25
Small meal 19 23 21
Sp = 2.625
S1 = 1.5
S2 = 1.732 =
24 – 21
0.7638 =
2.625
2.625 9 9 69

70. Hypothesis testing
Step 5: Make decision whether or not to reject null hypothesis
Observed t =
Critical t =
3.928 is farther out on the curve than 2.120
so, we do reject the null hypothesis
t(16) = 3.928; p < 0.05
Step 6: Conclusion: There appears to be a difference
in hearing sensitivity between the two groups 70

71. How to report the findings for a t-test
Mean of small meal was 21
How to report the findings for a t-test
Mean of big meal was 24
One paragraph summary of this study. Describe the IV & DV.
Present the two means, which type of test was conducted, and
the statistical results.
Start summary with two means (based on DV) for two levels of the IV
Observed t = 3.928
Finish with statistical summary t(9) = 3.93; p < 0.05
Describe type of test (z-test versus t-test) with brief overview of results
df = 16
p < 0.05
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic
We compared test scores for large and small meals. The mean test
score for the big meal was 24, and was 21 for the small meal.
A t-test was calculated and there was a significant difference in
test scores between the two types of meals t(16) = 3.928; p < 0.05 71

72. Let’s run more subjects using our excel!72

73. Let’s run more subjects using our excel!
Finding Means
Finding Means 73

74. Let’s run more subjects using our excel!
Finding degrees of freedom
Finding degrees of freedom 74

75. Let’s run more subjects using our excel!
Finding Observed t 75

76. Let’s run more subjects using our excel!
Finding Critical t 76

77. Let’s run more subjects using our excel!
Finding p value (Is it less than .05?) 77

78. What happened? We ran more subjects: Increased n
So, we decreased variability
Easier to find effect significant
even though effect size didn’t change
This is the sample size
This is the sample size
Small sample
Big sample 78

79. What happened? We ran more subjects: Increased n
So, we decreased variability
Easier to find effect significant
even though effect size didn’t change
This is variance for each sample
(Remember, variance is just standard deviation squared)
This is variance for each sample
(Remember, variance is just standard deviation squared)
Small sample
Big sample 79

80. Thank you!
See you next time!!