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Physics- atmospheric Sciences (PAS) - Room 201

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

4. Schedule of readings Before our next exam (November 10th)
OpenStax Chapters 1 – 10 and Chapter 13
Plous (2, 3, & 4)
Chapter 2: Cognitive Dissonance
Chapter 3: Memory and Hindsight Bias
Chapter 4: Context Dependence

5. Homework On class website:
Please print and complete homework worksheet #13
Hypothesis Testing using two-sample t-tests
Due: Tuesday November 3rd

6. By the end of lecture today 10/29/15
Logic of hypothesis testing
Hypothesis testing with z-score and t-scores (one-sample)
Hypothesis testing with t-scores (two independent samples)
Constructing brief, complete summary statements

8. Homework Review 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)

9. 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

10. 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

11. 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

12. 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

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. 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)?
One or two tailed test?
Balance between Type I versus Type II error
Critical statistic (e.g. z or t or F or r) 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
Step 5: Conclusion - tie findings back in to research problem

17. Hypothesis testing Is this a single sample or two sample test?
Is it a z or a t test or an ANOVA?
Hypothesis testing
Is it a one-tail test or a two tail test?
Step 1: Identify the research problem
Did the sheriff keep her promise to change response times from the previous average of 30 minutes?
Step 2: Describe the null and alternative hypotheses
Ho: The response times are not changed
H1: The response times did change
As the new chief of police, I am going to change response times for traffic accidents. Before I started the average response time was 30 minutes 17

18. Hypothesis testing Two-tailed test n = 10 df = 9 alpha = 0.05
Gather the data:
We measured the time for police to respond to 10 accidents
Two-tailed test
One or two tailed test?
What is our sample size
What is size of our degrees of freedom?
What is our alpha
What is our critical t value?
n = 10 (df = 9)
Alpha = .05
Decision rule: critical t = 2.62
n = 10
df = 9
alpha = 0.05 18

19. Hypothesis testing Two-tailed test Alpha of 0.05
Gather the data:
We measured the time for police to respond to 10 accidents
One or two tailed test?
What is our sample size
What is size of our degrees of freedom?
What is our alpha
What is our critical t value?
Decision rule: critical t = 2.262
Two-tailed test
Alpha of 0.05
Critical t (9) = 2.262 19

20. Average time for response before 30 minutes
Step 3: Calculations:
Average time for response before 30 minutes
Average time for response after 24 minutes
Observed t =
Step 4: Make decision whether or not to reject null hypothesis
Observed t =
Critical t = 2.262
-2.71 is farther out on the curve than 2.262
so, we do not reject the null hypothesis
Step 5: Conclusion: There appears to be a significant difference
between the sheriff’s times and 30 minutes 20

21. Hypothesis testing:
Did the sheriff keep her promise to reduce response times to less than 30 minutes?
Start summary with two means (based on DV) for two levels of the IV
notice we are comparing a sample mean with a population mean:
single sample t-test
Finish with statistical summary t(9) = -5.71; p < 0.05
Describe type of test (t-test versus anova) with brief overview of results
Or if it had been different results that *were not* significant:
t(9) = -1.71; ns
The mean response time for following the sheriff’s new plan was
24 minutes, while the mean response time prior to the new plan
was 30 minutes. A t-test was completed and there appears to be
a significant difference in the response time following the
implementation of the new plan t(9) = -2.71; p < 0.05
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 21

22. 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)

23. 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

24. 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 24

25. α = .05 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
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
Step 3: Decision rule
α = .05
One tail or two tail test? 25

26. Notice: Two different ways to think about it
Hypothesis testing
Step 3: Decision rule
α = .05
n1 = 3; n2 = 3
Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1)
= (3 - 1) + (3 – 1) = 4
Degrees of freedom total (df total) = (n total - 2)
two tailed test
Notice: Two different ways to think about it
Critical t(4)
= 2.776 26

27. two tail test
α= .05
(df) = 4
Critical t(4)
= 2.776 27

28. 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
= 1.732 6
Notice: s2 = 3.0 1 2 1
Notice: Simple Average = 3.5
= 2.0 8
Notice: s2 = 4.0 2 2 2
S2pooled =
(n1 – 1) s12 + (n2 – 1) s22
n1 + n2 - 2
S2pooled =
(3 – 1) (1.732) 2 + (3 – 1) (2)2
= 3.5 28

29. 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
S2p = 3.5 =
24 – 21
1.5275
= 1.964
3.5
3.5 3 3
Observed t 29

30. Hypothesis testing
Step 5: Make decision whether or not to reject null hypothesis
Observed t =
Critical t = 2.776
1.964 is not farther out on the curve than 2.776
so, we do not reject the null hypothesis
t(4) = 1.964; n.s.
Step 6: Conclusion: There appears to be no difference
in test scores between the two groups 30

31. 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 31

32. Figuring out the two means using Excel
Participant 1 2 3
Big Meal 22 25
Small meal 19 23 21 32

33. Figuring out the two means
Participant 1 2 3
Big Meal 22 25
Small meal 19 23 21 33

34. Figuring out the two means
Participant 1 2 3
Big Meal 22 25
Small meal 19 23 21
Mean= 21
Mean= 24 34

35. May want to remove means so don’t include in the t-test
Complete a t-test
Mean= 24
Participant 1 2 3
Big Meal 22 25
Small meal 19 23 21
May want to remove means so don’t include in the t-test 35

36. 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 36

37. Finding Means
Finding Means 37

38. Finding degrees of freedom38

39. Finding Observed t 39

40. Finding Critical t 40

41. Finding p value (Is it less than .05?)41

42. 42

43. 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 43

44. Thank you!
See you next time!!