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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 due – Thursday (October 29th)
On class website:
Please print and complete homework worksheet #12
Hypothesis testing with z-tests and t-tests

6. When: Monday evening October 19th
Stats Review for Exam 2
by Jonathon & Nick
When: Monday evening October 19th
6:30 – 7:30pm
Room: ILC 120
Cost: $5.00
Which of the following best describes your experience with the review session?
a. It was a very helpful review session
b. It was only okay
c. It was not a helpful review session
d. I did not attend this review

7. By the end of lecture today 10/27/15
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 z-score and t-scores (one-sample)
Hypothesis testing with t-scores (two independent samples)
Constructing brief, complete summary statements

9. It went really well! Exam 2 – Thanks for your patience and cooperation
The grades are posted

10. Frequency Score on Exam Remember… In a negatively skewed distribution:
mean < median < mode
Mode
87.5 = mode = tallest point
85 = median = middle score
82 = mean = balance point
Frequency
Score on Exam
Mean
Median
Note:
Always “frequency”
Note:
Label and Numbers

11. 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

12. We lose one degree of freedom for every parameter we estimate
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.

13. 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)

14. 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)

15. 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)

16. 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

17. 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)

18. 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

19. 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

20. 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

21. 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”

22. 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

23. 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)

24. 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

25. 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)

26. 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

27. 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

28. 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

29. 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.

30. These would be helpful to know by heart – please memorize
Pop Quiz – Part 1
Standard deviation and Variance
For Sample and Population
These would be helpful to know by heart – please memorize
these formula

31. Pop Quiz – Part 1
Standard deviation and Variance
For Sample and Population
Part 2:
When we move from a two-tailed test to a one-tailed test what happens to the critical z score (bigger or smaller?) - Draw a picture -
What affect does this have on the hypothesis test (easier or harder to reject the null?)

32. Pop Quiz – Part 3
1.  When do we use a t-test and when do we use a z-test?
        (Be sure to write out the formulae)
2.  How many steps in hypothesis testing
  (What are they?)
3.  What is our formula for degrees of freedom in one sample t-test?
4.  We lose one degree of freedom for every ________________
5.  What are the three parts to the summary (below)
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
no significant difference in the response time following the
implementation of the new plan t(9) = -1.71; n.s. 32

33. Pop Quiz
Standard deviation and Variance
For Sample and Population
Critical value gets smaller
Part 2:
When we move from a two-tailed test to a one-tailed test what happens to the critical z score (bigger or smaller?) - Draw a picture -
What affect does this have on the hypothesis test (easier or harder to reject the null?)
Gets easier to reject the null

34. Pop Quiz Writing Assignment
1.  When do we use a t-test and when do we use a z-test?
        (Be sure to write out the formulae)
Use the t-test when you don’t know the standard deviation of the population, and therefore have to estimate it using the standard deviation of the sample
Population versus sample standard deviation
Population versus sample standard deviation 34

35. Five steps to hypothesis testing
Step 1: Identify the research problem (hypothesis)
How is a t score similar to a z score?
How is a t score different than a z score?
Describe the null and alternative hypotheses
Same logic
and same steps
Step 2: Decision rule: find “critical z” score
Alpha level? (α = .05 or .01)?
One versus two-tailed test
Step 3: Calculate observed z score
Step 4: Compare “observed z” with “critical z”
If observed z > critical z then reject null
p < 0.05 and we have significant finding
Step 5: Conclusion - tie findings back in to research problem

36. Writing Assignment
3.  What is our formula for degrees of freedom in one sample t-test?
One sample t-test
Degrees of freedom =(df) = (n – 1)
First
Sample
Second
Sample
Two sample t-test
Degrees of freedom (df ) = (n1 - 1) + (n2 – 1)
4.  We lose one degree of freedom for every
parameter we estimate
Use the word "parameter” when describing a whole population (not just a sample). Usually we don’t know about the whole population so we have guess by using what we know about our sample.
A short-hand way to let the reader know it we are describing a population (a parameter) is to use a Greek letter – for example, σ for populations standard deviation, and an s for the sample.
In a t-test we never know the population standard deviation (parameter σ) we have to estimate this one parameter (using “s”), so we lose one df
our degree of freedom is n-1
Sample
standard deviation
Parameter: Population
standard deviation 36

37. Type of test with degrees of freedom Value of observed statistic
Writing Assignment
5.  What are the three parts to the summary (below)
Finish with statistical summary t(4) = 1.96; ns
Start summary with two means (based on DV) for two levels of the IV
Or if it *were* significant:
t(9) = 3.93; p < 0.05
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
no significant difference in the response time following the
implementation of the new plan t(9) = -1.71; n.s.
Describe type of test (t-test versus anova) with brief overview of results
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 37

39. Hypothesis testing: one sample t-test
Let’s jump right in
and do a t-test
Hypothesis testing: one sample t-test
Is the mean of my observed sample consistent with the known population mean or did it come from some other distribution?
We are given the following problem:
800 students took a chemistry exam. Accidentally, 25 students got an additional ten minutes. Did this extra time make a significant difference in the scores? The average number correct by the large class was 74.
The scores for the sample of 25 was
Please note:
In this example we are comparing our sample mean with the population mean
(One-sample t-test)
76, 72, 78, 80, 73
70, 81, 75, 79, 76
77, 79, 81, 74, 62
95, 81, 69, 84, 76
75, 77, 74, 72, 75

40. µ = 74 µ Hypothesis testing
Step 1: Identify the research problem / hypothesis
Did the extra time given to this sample of students affect
their chemistry test scores
Describe the null and alternative hypotheses
One tail or two tail test?
Ho:
µ = 74
= 74
H1:

41. We use a different table for t-tests
Hypothesis testing
Step 2: Decision rule
= .05
n = 25
Degrees of freedom (df) = (n - 1)
= (25 - 1) = 24
two tail test
This was for z scores
We use a different table for t-tests

42. two tail test
α= .05
(df) = 24
Critical t(24)
= 2.064

43. µ = 74 Hypothesis testing = = 868.16 = 6.01 24 x (x - x) (x - x)276 72 78 80 73 70 81 75 79 77 74 62 95 69 84
76 – 76.44
72 – 76.44
78 – 76.44
80 – 76.44
73 – 76.44
70 – 76.44
81 – 76.44
75 – 76.44
79 – 76.44
77 – 76.44
74 – 76.44
62 – 76.44
95 – 76.44
69 – 76.44
84 – 76.44
= -0.44 = = = = = = = = = = = = = = =
0.1936
2.4336
2.0736
6.5536
0.3136
5.9536
Step 3: Calculations
µ = 74 Σx = N
1911 25 =
= 76.44
N = 25
= 6.01
868.16 24
Σx = 1911
Σ(x- x) = 0
Σ(x- x)2 =

44. µ = 74 Hypothesis testing = 76.44 - 74 1.20 2.03 .
Step 3: Calculations
µ = 74
= 76.44
N = 25
s = 6.01 =
1.20
2.03
critical t
6.01 25
Observed t(24) = 2.03

45. Hypothesis testing
Step 4: Make decision whether or not to reject null hypothesis
Observed t(24) = 2.03
Critical t(24) = 2.064
2.03 is not farther out on the curve than 2.064,
so, we do not reject the null hypothesis
Step 6: Conclusion: The extra time did not have a significant
effect on the scores

46. Hypothesis testing:
Did the extra time given to these 25 students affect their average test score?
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(24) = 2.03; ns
Describe type of test (t-test versus z-test) with brief overview of results
Or if it had been different results that *were* significant: t(24) = -5.71; p < 0.05
The mean score for those students who where given extra time
was percent correct, while the mean score for the rest of
the class was only 74 percent correct. A t-test was completed
and there appears to be no significant difference in the test scores
for these two groups t(24) = 2.03; 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 46

47. Thank you!
See you next time!!