1. Introduction to Statistics for the Social Sciences SBS200, COMM200, GEOG200, PA200, POL200, or SOC200 Lecture Section 001, Spring 2016 Room 150 Harvill Building 9:00 - 9:50 Mondays, Wednesdays & Fridays
Welcome

3. By the end of lecture today 3/23/16
One-sample t-tests
Two sample t-test
Standard deviation and variance
Pooled Variance

4. Before next exam (April 8th)
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
On class website:
Please complete homework worksheet #20
Two-sample t hypothesis tests
Due: Friday, March 25th

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

8. 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

9. 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

10. 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

11. 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:

12. 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

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

14. 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 =

15. 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

16. 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

17. 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 17

18. 26.08 < µ < 33.92 mean + z σ = 30 ± (1.96)(2)
95%
< µ < 33.92
mean + z σ = 30 ± (1.96)(2)
99%
< µ < 35.16
mean + z σ = 30 ± (2.58)(2)

19. Melvin
Melvin
Mark
Difference not due sample size because both samples same size
Difference not due population variability because same population
Yes! Difference is due to sloppiness and random error in Melvin’s sample
Melvin

20. 6 – 5 = 4.0 .25 Two tailed test 1.96 (α = .05) 1 1 = = .25 16 4 √ 4.0
z- score : because we know the population standard deviation
Ho: µ = 5
Bags of potatoes from that plant are not different from other plants
Ha: µ ≠ 5
Bags of potatoes from that plant are different from other plants
Two tailed test
1.96
(α = .05) 1 1 =
= .25
6 – 5 16 4 √
= 4.0
.25
4.0
-1.96
1.96

21. Because the observed z (4.0 ) is bigger than critical z (1.96)
These three will always match
Yes
Yes
Probability of Type I error is always equal to alpha
Yes
.05
1.64 No
Because observed z (4.0) is still bigger than critical z (1.64)
2.58 No
Because observed z (4.0) is still bigger than critical z(2.58)
there is a difference
there is not
there is no difference
there is
1.96
2.58

22. 89 - 85 Two tailed test (α = .05) n – 1 =16 – 1 = 15
-2.13
2.13
t- score : because we don’t know the population standard deviation
Two tailed test
(α = .05)
n – 1 =16 – 1 = 15
Critical t(15) = 2.131
2.667 6 √ 16

23. Because the observed z (2.67) is bigger than critical z (2.13)
These three will always match
Yes
Yes
Probability of Type I error is always equal to alpha
Yes
.05
1.753 No
Because observed t (2.67) is still bigger than critical t (1.753)
2.947
Yes
Because observed t (2.67) is not bigger than critical t(2.947) No
These three will always match No No
consultant did improve morale
she did not
consultant did not improve morale
she did
2.131
2.947

24. Value of observed statistic
Finish with statistical summary z = 4.0; p < 0.05
Or if it *were not* significant:
z = 1.2 ; n.s.
Start summary with two means (based on DV) for two levels of the IV
Describe type of test (z-test versus t-test) with brief overview of results
n.s. = “not significant”
p<0.05 = “significant”
The average weight of bags of potatoes from this particular plant
is 6 pounds, while the average weight for population is 5 pounds.
A z-test was completed and this difference was found to be
statistically significant. We should fix the plant. (z = 4.0; p<0.05)
Value of observed statistic

25. 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 statistic (e.g. z or t) 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

26. Hypothesis testing with t-tests
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

27. Independent samples t-test
Are the two means significantly different from each other, or is the difference just due to chance?
Independent samples t-test
Donald is a consultant and leads training sessions. As part of his training sessions, he provides the students with breakfast. He has noticed that when he provides a full breakfast people seem to learn better than when he provides just a small meal (donuts and muffins). So, he put his hunch to the test. He had two classes, both with three people enrolled. The one group was given a big meal and the other group was given only a small meal. He then compared their test performance at the end of the day. Please test with an alpha = .05
Big Meal 22 25
Small meal 19 23 21
Mean= 21
Mean= 24
Got to figure this part out: We want to average from 2 samples - Call it “pooled”
x1 – x2
t =
24 – 21
variability
t =
variability 27

28. α = .05 Independent samples t-test
Step 1: Identify the research problem
Did the size of the meal affect the learning / test scores?
Step 2: Describe the null and alternative hypotheses
Step 3: Decision rule
α = .05
Two tailed test
n1 = 3; n2 = 3
Degrees of freedom total (df total) = (n1 - 1) + (n2 – 1)
= (3 - 1) + (3 – 1) = 4
Critical t(4)
= 2.776
Step 4: Calculate observed t score 28

29. Notice: Simple Average = 3.5
Mean= 21
Mean= 24
Big Meal
Deviation
From mean -2 1
Small Meal
Deviation
From mean -2 2
Squared
deviation 4 1
Squared Deviation 4
Big Meal 22 25
Small meal 19 23 21
Σ = 6
Σ = 8 6 3
Notice: s2 = 3.0 1 2 1
Notice: Simple Average = 3.5 8 4
Notice: s2 = 4.0 2 2 2
S2pooled =
(n1 – 1) s12 + (n2 – 1) s22
n1 + n2 - 2
S2pooled =
(3 – 1) (3) + (3 – 1) (4)
= 3.5 29

30. S2p = 3.5
Mean= 21
Mean= 24
Big Meal
Deviation
From mean -2 1
Small Meal
Deviation
From mean -2 2
Squared
deviation 4 1
Squared Deviation 4
Participant 1 2 3
Big Meal 22 25
Small meal 19 23 21
Σ = 6
Σ = 8 =
24 – 21
1.5275
= 1.964
3.5
3.5 3 3
Observed t
Observed t =
Critical t = 2.776
1.964 is not larger than so, we do not reject the null hypothesis
t(4) = 1.964; n.s.
Conclusion: There appears to be no difference between the groups 30

31. Type of test with degrees of freedom Value of observed statistic
We compared test scores for large and small meals. The mean test
scores for the big meal was 24, and was 21 for the small meal.
A t-test was calculated and there appears to be no significant
difference in test scores between the two types of meals,
t(4) = 1.964; n.s.
Type of test with degrees of freedom
n.s. = “not significant”
p<0.05 = “significant”
Value of observed statistic
Start summary with two means (based on DV) for two levels of the IV
Finish with statistical summary t(4) = 1.96; ns
Describe type of test (t-test versus anova) with brief overview of results
Or if it *were* significant:
t(9) = 3.93; p < 0.05 31

32. Homework Assignment
Using Excel ?

33. Homework Assignment
Using Excel

34. Thank you!
See you next time!!