1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2017 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays.
Welcome

2. A note on doodling

3. By the end of lecture today 3/22/17
Hypothesis testing with z scores
Hypothesis testing with t-tests
Interpreting
Alpha levels and Type I Errors
p values

4. Before next exam (April 7th)
Please read chapters in OpenStax textbook
Please read Chapters 2, 3, and 4 in Plous
Chapter 2: Cognitive Dissonance
Chapter 3: Memory and Hindsight Bias
Chapter 4: Context Dependence

5. Homework Assignment 19 Please complete this homework worksheet
One-sample z and t hypothesis test
Due: Friday, March 24th

6. Lab sessions Everyone will want to be enrolled
in one of the lab sessions
Project 3
this week

9. Preview of homework assignment

10. Preview of homework assignment

11. Preview of homework assignment

12. 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
1.64
95%
95% 5%
2.5%
2.5%
How would the critical z change?
Review

13. 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
Review

14. 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
Review

15. One versus two tail test of significance 5% versus 1% alpha levels
What if our observed z = ?
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
Review

16. We are looking to compare two means
Study Type 2: t-test
Comparing Two Means?
Use a t-test
We are looking to compare two means

17. We are looking to compare two means
Study Type 2: t-test
Comparing Two Means?
Use a t-test
We are looking to compare two means

18. 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

19. 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.

20. 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)

21. 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

22. 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)

23. 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)

24. 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)

25. Hypothesis testing: A review.
Difference between means
Hypothesis testing: A review
Variability of curve(s)
If the observed stat is more extreme than the critical stat in the distribution (curve):
then it is so rare, (taking into account the variability) we conclude it must be from some other distribution
decision considers effect size and variability
then we reject the null hypothesis – we have a significant result
then we have support for our alternative hypothesis
p < (p < α)
If the observed stat is NOT more extreme than the critical stat in the distribution (curve):
then we know it is a common score (either because the effect size is too small or because the variability is to big) and is likely to be part of this null distribution,
we conclude it must be from this distribution
decision considers effect size and variability – could be overly variable
then we do not reject the null hypothesis
then we do not have support for our alternative hypothesis
p not less than (p not less than α) p is n.s.
Difference between means
critical statistic
critical statistic
Variability of curve(s)
(within group variability)
Variability of curve(s)
Review

26. 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

27. 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

28. 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

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
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)

32. 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

33. 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)

34. 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

35. 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

36. 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

37. Thank you!
See you next time!!