1. Introduction to Hypothesis Testing
The One-Sample z Test

2. The One-Sample z Test Conditions of Applicability:
One group of subjects
Comparing to population with known mean and variance.
Note: this is not a common situation in Psychology!
PSYC 6130, PROF. J. ELDER

3. Example: Finish times for the 2005 Toronto Marathon (Oct 16, 2005)
Suppose your population of interest are women who ran the marathon (slightly artificial).
You hypothesize that women in their early twenties (20-24) are faster than the average woman who ran the marathon.
Here the ‘treatment’ is ‘youth’.
PSYC 6130, PROF. J. ELDER

4. Null Hypothesis Testing
Largely due to English mathematician Sir R.A. Fisher ( )
‘Proof by contradiction’
Suppose the null hypothesis is true
In our example, the null hypothesis is that the finishing times for young women are drawn from the same distribution as for the rest of the female contestants.
Knowing the mean and standard deviation of the population, we can compute the sampling distribution of the mean for a sample of size n. This is the null hypothesis distribution.
The mean time for our sample of young women should be plausible under this sampling distribution.
If it is not plausible, it suggests that the null hypothesis is false.
This lends credence to our alternate hypothesis (that young women are faster).
PSYC 6130, PROF. J. ELDER

5. How do we judge the plausibility of the null hypothesis?
The sample mean should be plausible under the sampling distribution of the mean.
Implausible
Fairly plausible
Highly plausible
PSYC 6130, PROF. J. ELDER

6. Plausibility of the null hypothesis
The plausibility of the null hypothesis is judged by computing the probability p of observing a sample mean that is at least as deviant from the population mean as the value we have observed.
PSYC 6130, PROF. J. ELDER

7. Plausibility of the null hypothesis
This computation is simplified by converting to z-scores.
Under the assumption of normality, we can determine this probability from a standard normal table.
PSYC 6130, PROF. J. ELDER

8. Results for 2005 Toronto Marathon
PSYC 6130, PROF. J. ELDER

9. Results for Random Sample of Women Under 25
PSYC 6130, PROF. J. ELDER

10. Statistical Decisions
We now know the probability that an observation like ours could have been drawn from the general female contestant population, i.e. that our ‘treatment of youth’ had no effect.
This probability is pretty small. Should we reject the null hypothesis? This is the process of turning a continuous probability (a real number) into a binary decision (yes or no).
If we reject the null hypothesis, there is a chance we will be wrong. We have to decide what chance we are willing to take, i.e. the maximum p-value we will accept as grounds for rejecting the null hypothesis.
We call this probability threshold the alpha (a) level. A typical value is .05.
The a-level must be decided prior to the experiment.
PSYC 6130, PROF. J. ELDER

11. Type I and Type II Errors
Type I Error: the null hypothesis is true and we reject it.
Type II Error: the null hypothesis is false and we fail to reject it.
Actual Situation
Researcher’s Decision
Null Hypothesis is True
Null Hypothesis is False
Accept the Null Hypothesis
Reject the Null Hypothesis
PSYC 6130, PROF. J. ELDER

12. Type I and Type II Errors
Which is more serious?
Type I can be bad, as rejecting the null hypothesis (e.g., ‘This stuff really works’), may cause actions to be taken that have no value.
Type II may not be so bad, if it is understood that the treatment may still have an effect (we fail to reject the null hypothesis, but we do not reject the alternate hypothesis).
But Type II may be bad if it leads to inaction when action would have produced good results (e.g., a cure for cancer).
PSYC 6130, PROF. J. ELDER

13. One-Tailed vs Two-Tailed Tests
Our marathon hypothesis was one-tailed, because we made a specific prediction about the direction of the effect (young women are faster).
Suppose we had simply hypothesized that young women are different.
PSYC 6130, PROF. J. ELDER

14. Two-Tailed Test
PSYC 6130, PROF. J. ELDER

15. One-Tailed vs Two-Tailed Tests
Use a one-tailed test when you have a specific reason to believe the effect will be in a particular direction, and you do not care if the effect is in the opposite direction.
Otherwise, use a two-tailed test.
One-tailed tests will always result in smaller p values, and hence a greater chance of reaching significance for your directional hypothesis.
The decision of whether to perform one-tailed or two-tailed tests must be made prior to data collection.
PSYC 6130, PROF. J. ELDER

16. Basic Procedure for Statistical Inference
State the hypothesis
Select the statistical test and significance level
Select the sample and collect the data
Find the region of rejection
Calculate the test statistic
Make the statistical decision
PSYC 6130, PROF. J. ELDER

17. Step 1. State the Hypothesis
PSYC 6130, PROF. J. ELDER

18. Step 2. Select the Statistical Test and the Significance Level
We are comparing a sample mean to a population with known mean and standard deviation  z-test
p=.05 is probably appropriate.
PSYC 6130, PROF. J. ELDER

19. Step 3. Select the Sample and Collect the Data
Ideally, we would randomly assign the treatment to a random sample of the population (Toronto Marathon women). Is this possible?
Instead, we randomly sample female contestants under 25.
PSYC 6130, PROF. J. ELDER

20. Step 4. Find the Region of Rejection
The z value defining the rejection region is called the critical value for your test, and is a function of the selected α-level. For this reason, we often denote the critical value as zα
PSYC 6130, PROF. J. ELDER

21. Step 5. Calculate the Test Statistic
PSYC 6130, PROF. J. ELDER

22. Step 6. Make the Statistical Decision
p<a: Reject null hypothesis.
p>a: Fail to reject null hypothesis.
PSYC 6130, PROF. J. ELDER

23. Example: Height of Female Psychology Graduate Students
Sample: Female students enrolled in PSYC 6130C
PSYC 6130, PROF. J. ELDER

24. Assumptions Underlying One-Sample z Test
Random sampling
Variable is normal
CLT: Deviations from normality ok as long as sample is large.
Dispersion of sampled population is the same as for the comparison population
e.g. suppose means are the same, but dispersion of sampled population is greater than dispersion of comparison population.
PSYC 6130, PROF. J. ELDER

25. Limitations of the One-Sample Test
Strongly depends on random sampling.
Better to have two groups of subjects: test (treatment) group and control group.
Problem of random sampling reduces to problem of random assignment to two groups: much easier!
PSYC 6130, PROF. J. ELDER

26. Reporting your results
Express your result in evocative English, then include the required numbers.
Follow APA style.
Example:
Young female runners were not found to be significantly faster than the general female contestant population, z=-1.31, p=0.095, one-tailed.
PSYC 6130, PROF. J. ELDER

27. More on Type I and Type II Errors
Total number of significant results
Consistent use of a fixed alpha-level determines the proportion of null experiments that generate significant results.
Don’t have enough information to know how many reported results are errors, because:
Don’t know the relative proportion of cases where H0 is true and H0 is false.
Don’t know the power of effective experiments.
Typically only significant results are reported (publication bias).
PSYC 6130, PROF. J. ELDER