1. INTEGRATED LEARNING CENTER
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INTEGRATED LEARNING CENTER
ILC 120
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2. BNAD 276: Statistical Inference in Management Spring 2016
Welcome
Green sheets

3. Schedule of readings Before our next exam (March 22nd)
OpenStax Chapters 1 – 11
Plous (10, 11, 12 & 14)
Chapter 10: The Representativeness Heuristic
Chapter 11: The Availability Heuristic
Chapter 12: Probability and Risk
Chapter 14: The Perception of Randomness

4. Homework
On class website:
Please print and complete homework #11 and #12
Due Thursday
Hypothesis Testing and Confidence Intervals

5. By the end of lecture today 3/1/16
Confidence Intervals
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?
One-tail versus Two-tail test
Type I versus Type II Errors

6. Confidence Interval of 95% Has and alpha of 5% α = .05
Critical z
-2.58
Critical z
2.58
Confidence Interval of 99%
Has and alpha of 1%
α = .01
99%
Critical z separates rare from common scores
Critical z
-1.96
Critical z
1.96
Confidence Interval of 95% Has and alpha of 5% α = .05
95%
Area associated with most extreme scores is called alpha
Critical z
-1.64
Critical z
1.64
Confidence Interval of 90%
Has and alpha of 10%
α = . 10
90%
Area in the tails is called alpha

7. 99% 95% 90% Moving from descriptive stats into inferential stats….
Area outside confidence interval is alpha
Area outside confidence interval is alpha
Moving from descriptive stats
into inferential stats….
99%
Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there
95%
Measurements that occur outside this middle ranges are suspicious, may be an error
or belong elsewhere
90%

8. Hypothesis testing: How do we know if something is going on
Hypothesis testing: How do we know if something is going on? How rare/weird is rare/weird enough?
Every day examples about when is weird, weird enough to
think something is going on?
Handing in blue versus white test forms
Psychic friend – guesses 999 out of 1,000 coin tosses right
Cancer clusters – how many cases before investigation
Weight gain treatment – one group gained an average of pound more than other group…what if 10?

9. Why do we care about the z scores that define the middle 95% of the curve? Inferential Statistics
Hypothesis testing with z scores allows us to make inferences
about whether the sample mean is consistent with the known
population mean.
Is the mean of my observed sample consistent with the
known population mean or did it come from some other
distribution?

10. Why do we care about the z scores that define the middle 95% of the curve?
If the z score falls outside the middle
95% of the curve, it must be from
some other distribution
Main assumption: We assume that
weird, or unusual or rare things
don’t happen
If a score falls out into the 5% range we conclude that it “must be”
actually a common score but from some other distribution
That’s why we care about the z scores that define the middle
95% of the curve

11. I’m not an outlier I just haven’t found my distribution yet.
Main assumption: We assume that weird, or unusual or rare things don’t happen
I’m not an outlier I just haven’t found my distribution yet
If a score falls out into the tails (low probability) we
conclude that it “must be” a common score from some
other distribution

12. .. 95% 95% X X Reject the null hypothesis
Relative to this distribution I am unusual maybe even an outlier X
95% X
Relative to this distribution I am utterly typical
Do not reject the
null hypothesis

13. Rejecting the null hypothesis.
null
notnull
big z score x x
If the observed z falls beyond the critical z in the distribution (curve):
then it is so rare, we conclude it must be from some other distribution
then we reject the null hypothesis
then we have support for our alternative hypothesis
Alternative Hypothesis
If the observed z falls within the critical z in the distribution (curve):
then we know it is a common score and is likely to be part of this distribution,
we conclude it must be from this distribution
then we do not reject the null hypothesis
then we do not have support for our alternative .
null x x
small z score

14. I’m not an outlier I just haven’t found my distribution yet.
Main assumption: We assume that weird, or unusual or rare things don’t happen
I’m not an outlier I just haven’t found my distribution yet
If a score falls out into the tails (low probability) we
conclude that it “must be” a common score from some
other distribution

15. 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)?
Critical z value?
Step 3: Calculations from collected data – “observed z”
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

16. Confidence Interval of 95% Has and alpha of 5% α = .05
Critical z
-2.58
Critical z
2.58
Confidence Interval of 99%
Has and alpha of 1%
α = .01
99%
Area in the tails is called alpha
Critical z
-1.96
Critical z
1.96
Confidence Interval of 95% Has and alpha of 5% α = .05
95%
Critical Z separates rare from common scores
Critical z
-1.64
Critical z
1.64
Confidence Interval of 90%
Has and alpha of 10%
α = . 10
90%
It would be easiest to reject the null at which alpha level?
why?

17. Deciding whether or not to reject the null hypothesis. 05 versus
Deciding whether or not to reject the null hypothesis .05 versus .01 alpha levels
What if our observed z = 2.0?
How would the critical z change?
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.96 or
+1.96
p < 0.05
Yes, Significant difference
Reject the null
Remember,
reject the null if the observed z is bigger than the critical z
-2.58 or
+2.58
Not a Significant difference
Do not Reject the null

18. Rejecting the null hypothesis
The result is “statistically significant” if:
the observed statistic is larger than the critical statistic
observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x2) to be big!!
the p value is less than 0.05 (which is our alpha)
p < If we want to reject the null, we want our “p” to be small!!
we reject the null hypothesis
then we have support for our alternative hypothesis

19. Deciding whether or not to reject the null hypothesis. 05 versus
Deciding whether or not to reject the null hypothesis .05 versus .01 alpha levels
What if our observed z = 1.5?
How would the critical z change?
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.96 or
+1.96
Do Not
Reject the null
Not a Significant difference
Remember,
reject the null if the observed z is bigger than the critical z
-2.58 or
+2.58
Not a Significant difference
Do Not
Reject the null

20. Deciding whether or not to reject the null hypothesis. 05 versus
Deciding whether or not to reject the null hypothesis .05 versus .01 alpha levels
What if our observed z = -3.9?
How would the critical z change?
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.96 or
+1.96
p < 0.05
Yes, Significant difference
Reject the null
Remember,
reject the null if the observed z is bigger than the critical z
-2.58 or
+2.58
p < 0.01
Yes, Significant difference
Reject the null

21. Deciding whether or not to reject the null hypothesis. 05 versus
Deciding whether or not to reject the null hypothesis .05 versus .01 alpha levels
What if our observed z = -2.52?
How would the critical z change?
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.96 or
+1.96
p < 0.05
Yes, Significant difference
Reject the null
Remember,
reject the null if the observed z is bigger than the critical z
-2.58 or
+2.58
Not a Significant difference
Do not Reject the null

22. Setting our decision threshold
Area in the tails is alpha
99%
α = .01
95%
α = .05
90%
α = .10
Setting our decision threshold
Level of significance is called alpha (α)
The degree of rarity required for an observed outcome
to be “weird enough” to reject the null hypothesis
Which alpha level would be associated with most “weird” or rare scores?
Critical z: A z score that separates common from rare outcomes
and hence dictates whether the null hypothesis should
be retained (same logic will hold for “critical t”)
If the observed z falls beyond the critical z in the distribution
(curve) then it is so rare, we conclude it must be from some
other distribution

23. How would the critical z change?
One versus two tail test of significance: Comparing different critical scores (but same alpha level – e.g. alpha = 5%)
One versus two tailed test of significance
z score = 1.64
95%
95% 5%
2.5%
2.5%
How would the critical z change?
Pros and cons…

24. One versus two tail test of significance 5% versus 1% alpha levels
How would the critical z change?
One-tailed
Two-tailed
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01 1% 5%
2.5%
.5%
.5%
2.5%
-1.64 or
+1.64
-1.96 or
+1.96
-2.33 or
+2.33
-2.58 or
+2.58

25. One versus two tail test of significance 5% versus 1% alpha levels
What if our observed z = 2.0?
How would the critical z change?
One-tailed
Two-tailed
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.64 or
+1.64
-1.96 or
+1.96
Remember,
reject the null if the observed z is bigger than the critical z
Reject the null
Reject the null
-2.33 or
+2.33
-2.58 or
+2.58
Do not Reject the null
Do not Reject the null

26. One versus two tail test of significance 5% versus 1% alpha levels
What if our observed z = 1.75?
How would the critical z change?
One-tailed
Two-tailed
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.64 or
+1.64
-1.96 or
+1.96
Remember,
reject the null if the observed z is bigger than the critical z
Do not
Reject the null
Reject the null
-2.33 or
+2.33
-2.58 or
+2.58
Do not Reject the null
Do not Reject the null

27. One versus two tail test of significance 5% versus 1% alpha levels
What if our observed z = 2.45?
How would the critical z change?
One-tailed
Two-tailed
α = 0.05
Significance level = .05
α = 0.01
Significance level = .01
-1.64 or
+1.64
-1.96 or
+1.96
Remember,
reject the null if the observed z is bigger than the critical z
Reject the null
Reject the null
-2.33 or
+2.33
-2.58 or
+2.58
Reject the null
Do not Reject the null

28. 99% 95% 90% Area outside confidence interval is alpha
Logic of inferential stats….
Measurements that occur within the middle part of the curve are ordinary (typical) and probably belong there (Do not reject the null hypothesis)
99%
Measurements that occur outside this middle range are suspicious, may be an error
or belong elsewhere
(Do reject the null hypothesis)
95%
90%
Review

29. Rejecting the null hypothesis
The result is “statistically significant” if:
the observed statistic is larger than the critical statistic
observed stat > critical stat If we want to reject the null, we want our t (or z or r or F or x2) to be big!!
the p value is less than 0.05 (which is our alpha)
p < If we want to reject the null, we want our “p” to be small!!
we reject the null hypothesis
then we have support for our alternative hypothesis
A note on decision making following procedure
versus being right relative to the “TRUTH”
Review

30. Procedures versus outcome Best guess versus “truth”.
Decision making:
Procedures versus outcome
Best guess versus “truth”
What does it mean to be correct?
Why do we say:
“innocent until proven guilty”
“not guilty” rather than “innocent”
Is it possible we got a verdict wrong?

31. We make decisions at Security Check Points.
We make decisions at Security Check Points .

32. Does this airline passenger have a snow globe?.
Type I or Type II error? .
Does this airline passenger have a snow globe?
Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it???!!
As detectives, do we accuse her of brandishing a snow globe?

33. Does this airline passenger have a snow globe?.
Does this airline passenger have a snow globe?
Status of Null Hypothesis (actually, via magic truth-line)
Are we correct or have we made a Type I or Type II error?
True Ho
No snow globe
False Ho
Yes snow globe
You are wrong!
Type II error (miss)
Do not reject Ho “no snow globe move on”
You are right!
Correct decision
Decision made by experimenter
You are wrong!
Type I error (false alarm)
Reject Ho
“yes snow globe, stop!”
You are right!
Correct decision
Note: Null Hypothesis means she does not have a snow globe (that nothing unusual is happening) – Should we reject it???!!

34. Type I error (false alarm).
Type I or type II error?
Decision made by experimenter
Reject Ho
Do not Reject Ho
True Ho
False Ho
You are right!
Correct decision
You are wrong!
Type I error (false alarm)
Type II error (miss)
Does this airline passenger
have a snow globe?
Two ways to be correct:
Say she does have snow globe when she does have snow globe
Say she doesn’t have any when she doesn’t have any
Two ways to be incorrect:
Say she does when she doesn’t (false alarm)
Say she does not have any when she does (miss)
What would null hypothesis be?
This passenger does not have any snow globe
Type I error: Rejecting a true null hypothesis
Saying the she does have snow globe when in fact she does not (false alarm)
Type II error: Not rejecting a false null hypothesis
Saying she does not have snow globe when in fact she does (miss)

35. Thank you!
See you next time!!