1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Fall 2016 Room 150 Harvill Building 10: :50 Mondays, Wednesdays & Fridays.
Welcome

3. By the end of lecture today 10/31/16
Use this as your
study guide
Logic of hypothesis
testing with t-tests
Steps for hypothesis testing
for t-tests
Levels of significance (alpha)
what does alpha of .05 mean?
what does p < 0.05 mean?
what does alpha of .01 mean?
what does p < 0.01 mean?
Using Excel to complete t-tests

4. Please hand in homework assignment 17 now
Homework Assignments
Please hand in homework assignment 17 now
Homework Assignment 17:
Hypothesis Testing Type I versus Type II Errors
Due Monday October 31st
Homework Assignment 18
Hypothesis testing using z scores and t scores
Comparing Two means
(Single sample and population mean)
Due Wednesday November 2nd

5. Before next exam (November 18th)
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

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

11. The three parts to the summary
1. State independent and dependent variables and means for each group
2. Describe type of test, and whether significance was found
3. Finish with statistical summary F(df, df) = observed; p< 0.05 (or “n.s.”)
Start summary with means for each level of the IV
The average number of cookies sold for three different incentives
were compared. The mean number of cookie boxes sold for the
“Hawaii” incentive was 14 , the mean number of cookies boxes sold
for the “Bicycle” incentive was 12, and the mean number of cookies
sold for the “No” incentive was 10. An ANOVA was conducted and
there appears to be no significant difference in the number of
cookies sold as a result of the different levels of incentive
F(2, 12) = 2.73; n.s.
Describe type of test and results
n.s. = “not significant”
p < 0.05 = “significant”
Type Study
Degree of freedom
Observed
Score
Statistical Summary 11

13. Review 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
Review

14. 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
Review

15. 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
Review

16. Remember, you should know these four formulas by heart
“SS” = “Sum of Squares”
“SS” = “Sum of Squares”
“n” = number of scores
“SS” = “Sum of Squares”
“SS” = “Sum of Squares”
“SS” = “Sum of Squares”
“df” = degrees of freedom
“SS” = “Sum of Squares”
Remember, you should know
these four formulas by heart

17. Remember, you should know these four formulas by heart

18. 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.

19. Five steps to hypothesis testing
Step 1:Identify the research problem (hypothesis)
Describe the null and alternative hypotheses
Step 2: Decision rule: find “critical score”
z score:
Alpha level? (α = .05 vs .01)
Prediction (one vs two-tailed)
t score:
Alpha level? (α = .05 vs .01)
Prediction (one vs two-tailed)
Degrees of freedom
Population versus sample standard deviation
Population versus sample standard deviation
Step 3: Calculations
Step 4: Make decision - If calculated score > critical score then reject null
Step 5: Conclusion - tie findings back in to research problem
State IV, DV and means - Type of test and whether significant – Symbolic summary

20. 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

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
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

22. 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)

23. 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
Remember these
useful values for z-scores? .
1.64
1.96
2.58

24. 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)

25. 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)

26. 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

27. 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)

28. 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)

29. 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)

30. Thank you!
See you next time!!