1. Reasoning in Psychology Using Statistics
2017

2. Quiz 5
pushed back to due tomorrow (Tue Mar. 28) night at midnight
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3. Testing Hypotheses Hypothesis testing: a five step program
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data from your sample
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Testing Hypotheses

4. Testing Hypotheses Hypothesis testing: a five step program
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data from your sample
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Testing Hypotheses

5. Performing your statistical test
What are we doing when we test the hypotheses?
Consider a variation of our memory experiment example
Population of
memory patients
MemoryTest
μ & σ known
Memory
treatment
patients
Test X
Compare these
two means
We test this one
Conclusions:
H0: The memory treatment sample are the same as those in the population of memory patients.
HA: They are not the same as those in the population of memory patients
Performing your statistical test

6. Performing your statistical test
What are we doing when we test the hypotheses?
Real world (‘truth’)
H0: is true (no treatment effect)
H0: is false (is a treatment effect)
One population XA
H0: The memory treatment sample are the same as those in the population of memory patients.
HA: They are not the same as those in the population of memory patients
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Performing your statistical test

7. Performing your statistical test
What are we doing when we test the hypotheses?
Real world (‘truth’)
H0: is true (no treatment effect)
H0: is false (is a treatment effect)
One population
Two populations XA XA
H0: The memory treatment sample are the same as those in the population of memory patients.
HA: They are not the same as those in the population of memory patients
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Performing your statistical test

8. Performing your statistical test
What are we doing when we test the hypotheses?
Real world (‘truth’)
We test this one
H0: is true (no treatment effect)
H0: is false (is a treatment effect)
One population
Two populations XA XA
H0: The memory treatment sample are the same as those in the population of memory patients.
HA: They are not the same as those in the population of memory patients
Real world (‘truth’)
H0 is correct
H0 is wrong
Experimenter’s conclusions
Reject H0
Fail to Reject H0
Type I error
Type II error
Performing your statistical test

9. “Generic” statistical test
The generic test statistic distribution (a transformation of the distribution of sample means)
To reject the H0, you want a computed test statistics that is large
The probability of having a sample with that mean is very low
What is large enough?
The alpha level gives us the decision criterion
Distribution of the test statistic
α-level determines where these boundaries go
“Generic” statistical test

10. “Generic” statistical test
The generic test statistic distribution (a transformation of the distribution of sample means)
To reject the H0, you want a computed test statistics that is large
The probability of having a sample with that mean is very low
What is large enough?
The alpha level gives us the decision criterion
Distribution of the test statistic
If test statistic is here Reject H0
If test statistic is here Fail to reject H0
“Generic” statistical test

11. “Generic” statistical test
The alpha level gives us the decision criterion
Two -tailed z
.00
.01 …
.06 :
1.0
1.9
3.0
.5000
.1587
.0287
.0013
.4960
.1562
.0281
.0250
α = 0.05
0.025
split up
into the
two tails
Reject H0
Fail to reject H0
Go to the table (unit normal table for z-test) and find the z that has in the tails.
Zcritical = ±1.96
“Generic” statistical test

12. “Generic” statistical test
The alpha level gives us the decision criterion
Two -tailed
One -tailed
Reject H0
Fail to reject H0
α = 0.05
0.05
all of it in one tail
Reject H0
Reject H0
Fail to reject H0
Fail to reject H0
Go to the table (unit normal table for z-test) and find the z that has in the tail. Zcritical = z
.00 …
.05 :
1.6
0.5000
0.0548 ….
.4801
.0495
“Generic” statistical test

13. “Generic” statistical test
The alpha level gives us the decision criterion
Two -tailed
One -tailed
Reject H0
Fail to reject H0
α = 0.05
all of it in one tail
0.05
Reject H0
Reject H0
Fail to reject H0
Fail to reject H0
Go to the table (unit normal table for z-test) and find the z that has in the tail. Zcritical = z
.00 …
.05 :
1.6
0.5000
0.0548 ….
.4801
.0495
“Generic” statistical test

14. 1-sample z-test Population of memory patients Memory Test σ is known
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
Population of
memory patients
Memory Test
σ is known
μ is known
Memory
treatment
patients
Test X
Compare these
two means
1-sample z-test

15. 1-sample z-test 1 sample Population of memory patients Memory Test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
1 sample
Population of
memory patients
Memory Test
σ is known
μ is known
Memory
treatment
patients
Test X
Compare these
two means
1-sample z-test

16. 1-sample z-test 1 sample 1 score per subject Population of
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
1 sample
1 score per subject
Population of
memory patients
Memory Test
σ is known
μ is known
Memory
treatment
patients
Test X
Compare these
two means
1-sample z-test

17. 1-sample z-test 1-sample z-test 1 sample 1 score per subject
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
1 sample
1 score per subject
Population mean (μ) and standard deviation (σ) are known (assume Normal dist)
Population of
memory patients
Memory Test
σ is known
μ is known
Memory
treatment
patients
Test X
Compare these
two means
1-sample z-test
1-sample z-test

18. Performing your statistical test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
One -tailed
Step 1: State your hypotheses
μTreatment > μpop = 60
Step 2: Set your decision criteria
H0:
the memory treatment sample are the same as those in the population of memory patients (or even worse).
Step 3: Collect your data
Step 4: Compute your test statistics
HA:
the memory treatment sample perform better (fewer errors) than those in the population of memory patients
μTreatment < μpop = 60
Step 5: Make a decision about your null hypothesis
Performing your statistical test

19. Performing your statistical test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
H0:
μTreatment > μpop = 60
HA:
μTreatment < μpop = 60
One -tailed
α = 0.05
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Performing your statistical test

20. Performing your statistical test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
H0:
μTreatment > μpop = 60
HA:
μTreatment < μpop = 60
n = 16, X = 55
One -tailed
α = 0.05
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Performing your statistical test

21. Performing your statistical test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
H0:
μTreatment > μpop = 60
HA:
μTreatment < μpop = 60
One -tailed
α = 0.05
n = 16, X = 55
Step 1: State your hypotheses
= -2.5
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Performing your statistical test

22. Performing your statistical test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors (while the typical memory patient averages μ = 60 errors, with σ = 8). Test using α = 0.05.
H0:
μTreatment > μpop = 60
HA:
μTreatment < μpop = 60
One -tailed
α = 0.05
n = 16, X = 55
Step 1: State your hypotheses
= -2.5
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics 5%
Step 5: Make a decision about your null hypothesis
Reject H0
- Support for our HA, the evidence suggests that the treatment decreases the number of memory errors
Performing your statistical test

23. 1-sample t-test The 1-sample t-test
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors).
The 1-sample t-test
Population standard deviation (s) is NOT known
1-sample t-test

24. One sample t-test The 1-sample t-test Hypotheses: H0: HA:
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors).
The 1-sample t-test
Population of
memory patients
MemoryTest
σ is NOT known
μ is known
Memory
treatment
patients
Test X
Compare these
two means
Hypotheses:
The memory treatment sample are the same as those in the population of memory patients (or even worse), that is, μA ≥ μ0.
The memory treatment sample perform better (fewer errors) than those in the population of memory patients, that is, μA < μ0.
H0:
HA:
One sample t-test

25. Testing Hypotheses Hypothesis testing: a five step program
The 1-sample t-test
Hypothesis testing: a five step program
Step 1: State your hypotheses
Step 2: Set your decision criteria
Step 3: Collect your data
Step 4: Compute your test statistics
Step 5: Make a decision about your null hypothesis
Same
Different
Compute your estimated standard error
Compute your t-statistic
Compute your degrees of freedom
Testing Hypotheses

26. Based on standard error or an estimate of the standard error
What are we doing when we test the hypotheses?
Computing a test statistic: Generic test
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate of the standard error
Hypothesis testing

27. Based on standard error or an estimate of the standard error
What are we doing when we test the hypotheses?
Computing a test statistic: 1-sample t-test
Could be difference between a sample and a population, or between different samples
Based on standard error or an estimate of the standard error
1-sample t-test

28. Based onandard error or estimate of standard error
What are we doing when we test the hypotheses?
Computing a test statistic: 1-sample t-test
Could be difference between a sample and a population, or between different samples
Based onandard error or estimate of standard error
1-sample t-test

29. 1-sample z
1-sample t
identical
Test statistic
1-sample t-test

30. Use our best guess, sample standard deviation (s)
1-sample z
1-sample t
Test statistic
different
Diff. Expected by chance
Standard error
If don’t know this,
so need to estimate it
Use our best guess, sample standard deviation (s) XA s
1-sample t-test

31. Use our best guess, sample standard deviation (s)
1-sample z
1-sample t
Test statistic
different
Diff. Expected by chance
Standard error
If don’t know this,
so need to estimate it
Estimated standard error
Degrees of freedom
Use our best guess, sample standard deviation (s)
1-sample t-test

32. The t-statistic distribution (a transformation of the distribution of sample means)
To reject the H0, you want a computed test statistics that is large
The alpha level gives us the decision criterion
New table: the t-table; set up to find critical t-value for given p-value
Distribution of the t-statistic
If test statistic is beyond here Reject H0
If test statistic is here, fail to reject H0
Note: Bottom row: same as z. The t-distribution becomes the Normal distribution when df = ∞
The t-distribution

33. Each row corresponds to a different curve
The t distributions are like the z distribution (μ = 0; σ =1).
α levels
New table: the t-table
1-tailed
- or -
2-tailed
Degrees of freedom df
Critical values of t
tcrit
Each row corresponds to a different curve
The t-distribution

34. The t-distribution New table: the t-table
As df gets smaller, need larger tcrit, shown here for α = .05, 2-tailed. Curve flattens (becomes platykurdic).
df=
df =
df =
The t-distribution

35. What is the tcrit for a 2-tailed hypothesis test with a sample size of n = 6 and an α level of 0.05?
α = 0.05
2-tailed
df = n - 1 = 5
tcrit =
The t-distribution

36. What is the tcrit for a 1-tailed hypothesis test with a sample size of n = 6 and an α level of 0.05?
α = 0.05
n = 6
1-tailed
df = n - 1 = 5
tcrit =
1-tailed, larger critical region in 1-tail, so smaller critical value needed.
tcrit = vs. ±2.571
The t-distribution

37. 1-sample t-test An example: 1-sample t-test H0: HA:
Memory treatment sample same as or make more errors than population of memory patients.
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05.
1-tailed
H0: μA > μ0 = 60
HA:
Sample make fewer errors than population of memory patients
HA: μA < μ0 = 60
Step 1: Hypotheses
1-sample t-test

38. 1-sample t-test An example: 1-sample t-test H0: μA > μ0 = 60
HA: μA < μ0 = 60
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05.
1-tailed
α = 0.05
Step 1: Hypotheses
Step 2: Criterion for decision
1-sample t-test

39. 1-sample t-test An example: 1-sample t-test = 55, s = 8
H0: μA > μ0 = 60
HA: μA < μ0 = 60
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. . His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05.
1-tailed
α = 0.05
= 55, s = 8
Step 1: Hypotheses
Step 2: Criterion for decision
Step 3: Sample statistics
1-sample t-test

40. 1-sample t-test An example: 1-sample t-test = 55, s = 8
H0: μA > μ0 = 60
HA: μA < μ0 = 60
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. . His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05.
1-tailed
α = 0.05
= 55, s = 8
Step 1: Hypotheses
= -2.5
Step 2: Criterion for decision
Step 3: Sample statistics
Step 4: Test statistic
1-sample t-test

41. 1-sample t-test An example: 1-sample t-test = 55, s = 8 tcrit = -1.753
H0: μA > μ0 = 60
HA: μA < μ0 = 60
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. . His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05.
1-tailed
α = 0.05
= 55, s = 8
Step 1: Hypotheses
Step 2: Criterion for decision
Step 3: Sample statistics
Step 4: Test statistic
Step 5: Compare observed to critical value & make a decision about your null hypothesis
tcrit =
1-sample t-test

42. 1-sample t-test An example: 1-sample t-test = 55, s = 8 tobs=-2.5
H0: μA > μ0 = 60
HA: μA < μ0 = 60
Dr. Mnemonic develops a new treatment for patients with a memory disorder. He hypothesizes that the treatment will improve memory performance. To test it he collects a sample of 16 patients and gives them his new treatment. Following the treatment he gives them a standard memory test. His sample averaged 55 errors, with s = 8 (while for the typical memory patient μ = 60 errors). Test this with α level = 0.05.
1-tailed
α = 0.05
= 55, s = 8
tobs=-2.5
Step 1: Hypotheses
Step 2: Criterion for decision
Step 3: Sample statistics
Step 4: Test statistic
Step 5: Compare observed to critical value & make a decision about your null hypothesis
= tcrit
“Evidence supports the hypothesis that the memory treatment improved performance”
1-sample t-test

43. SPSS output for 1-sample t-test
= 8 – 10 = -5 .4
Cover later:
Estimate difference between -3 and -1
In SPSS, compare observed & critical p-values.
1-tail p = .0005
Less than α = .05?
SPSS output for 1-sample t-test

44. In lab: Practice using 1-sample t-tests and the hypothesis testing framework
Questions?
Wrap up