1. Introduction to Statistics for the Social Sciences SBS200 - Lecture Section 001, Spring 2019 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays.
March 18

2. Even if you have not yet registered your clicker you can still participate
The
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Sheets

3. Before next exam (April 5th)
Schedule of readings
Before next exam (April 5th)
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

4. Labs continue this week
Lab sessions
Everyone will want to be enrolled
in one of the lab sessions
Labs continue this week

6. Let’s do ANOVA Using Excel
A girlscout troop leader wondered whether providing an
incentive to whomever sold the most girlscout cookies would
have an effect on the number cookies sold. She provided
a big incentive to one troop (trip to Hawaii), a lesser
incentive to a second troop (bicycle), and no incentive to
a third group, and then looked to see who sold more cookies.
Troop 1
(Nada) 10 8 12 7 13
Troop 2
(bicycle) 12 14 10 11 13
Troop 3
(Hawaii) 14 9 19 13 15
n = 5
x = 10
n = 5
x = 12
n = 5
x = 14

7. Let’s do one Replication of study (new data)

8. Let’s do same problem Using MS Excel

9. Let’s do same problem Using MS Excel

10. Means for
Each group

11. No, so it is not significant Do not reject null
F critical (is observed F greater than critical F?)
P-value (is it less than .05?)

12. Make decision whether or not to reject null hypothesis
Observed F = 2.73
Critical F(2,12) = 3.89
2.7 is not farther out on the curve than 3.89
so, we do not reject the null hypothesis
Also p-value is not smaller than 0.05
so we do not reject the null hypothesis
Step 6: Conclusion: There appears to be no effect of type of
incentive on number of girl scout cookies sold

13. Make decision whether or not to reject null hypothesis
Observed F = 2.73
F(2,12) = 2.73; n.s.
Critical F(2,12) = 3.89
2.7 is not farther out on the curve than 3.89
so, we do not reject the null hypothesis
Conclusion: There appears to be no effect of type of
incentive on number of girl scout cookies sold
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.

17. Which is worse? Type I or type II error.
Which is worse?
Type I or type II error
What if we were looking to see if an individual were guilty of a crime?
Two ways to be correct:
Say they are guilty when they are guilty
Say they are not guilty when they are innocent
Two ways to be incorrect:
Say they are guilty when they are not
Say they are not guilty when they are
What would null hypothesis be?
This person is innocent - there is no crime here
Type I error: Rejecting a true null hypothesis
Saying the person is guilty when they are not (false alarm)
Sending an innocent person to jail (& guilty person to stays free)
Type II error: Not rejecting a false null hypothesis
Saying the person in innocent when they are guilty (miss)
Allowing a guilty person to stay free

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. 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. 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

24. 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

25. 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

26. 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

27. 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:

28. 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

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

30. 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 =

31. 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

32. 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

33. 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 33

35. Preview of homework assignment

36. Preview of homework assignment

37. Thank you!
See you next time!!