1. Hypothesis testing
March 20, 2000

2. Two-tailed tests
A two-tailed test means that an equal part of the standard error is in each “tail” of the distribution.

3. Confidence interval
Using z scores, we can calculate the actual values represented by the boundary of the area in the tails.
For example, if we want a 95% confidence limit, we find the values for which 2.5% of scores (half of 5%) fall below and 2.5% fall above.

4. confidence interval
Homework review

5. Hypothesis testing with sample data Do two variables have a nonrandom relationship?
Four basic steps
State the null and research hypothesis
Select a significance level (alpha ())
Select and compute a test statistic
Make a decision by comparing to critical value of the test statistic.

6. Example - t test Meier/Brudney, 11.12
Change in toll booths from human collection to machine collection.
Human attendants (population)
 = 1,253 cars per hour
New system; sample size of 100 hours
 = 1,261 cars per hour st dev = 59
State a hypothesis, null hypothesis, and evaluate them. What is your conclusion?
You are comparing the sample mean to the population mean.

7. Example - t test State the research and null hypothesis.
H1: The average number of cars passing through automated booths is higher than the average number of cars passing through booths with human attendants.
H0:There is no difference in the average number of cars passing through booths with automated collections and those with human attendant collection.
Note: this is a directional hypothesis. It states that one is higher or lower than the other. Use a one-tailed test.

8. Example - t test
Compute the statistic

9. Example - t test Find the critical t
Because this problem has a directional hypothesis, we will use a one-tailed test.
The degrees of freedom are equal to the sample size minus 1.
Using the t table, find the critical t for  = .05 and df = 99.
The critical t = 1.66

10. Example - t test So, the calculated t = 3.05
the critical t (t.05) = 1.66
Because the calculated t score is greater than the critical t, we reject the null hypothesis.
There is a statistically significant difference in the mean number of cars passing through the automated tool booths and the mean number of cars passing through booths with human attendants.

11. T-test
Practice problems

12. One sample z test
We use the t-test when we want to compare the means of a sample to a population or the means of two samples if we know the mean and standard deviation for the samples, but only the mean of the population.
If, however, we also know the standard deviation of the population, we use a one-sample z test.

13. One sample z test Don’t confuse with z scores.
One-sample z test is an inferential statistic.
Z score is a descriptive statistic.

14. Example - one sample z test
Over the course of many years, a Midwestern college has a mean law school acceptance rate of 50% per year. The standard deviation = 3.5
The pre-law advisor developed a program intended to improve the rate of acceptance in 1999.
In 1999, 130 graduates applied to law school; 73 were accepted (56%).
Did law school acceptances improve following the program?

15. Example - one sample z test
State the research and null hypotheses.
H1: Law school admission rates were higher for those participating in a pre-law advising program.
H0:There is no difference in law school admission rates between those who participate in an advising program and those who do not.
Note: This is a one-tailed hypothesis. One mean is higher than the other.

16. Example - one sample z test
Select an .
Remember, in social sciences the general standard is  = .05 (remember, this equates to what percent of the time you are likely to be wrong).
Choose a test statistic - since we are comparing the mean of a group (sample) to a known population mean and standard deviation, the one sample z test is an appropriate statistic.

17. Example - one sample z test
Compute the statistic. Note: the denominator is the formula for a standard error.

18. Example - one sample z test
Calculate z

19. Critical value of z (two-tailed test for non-directional Ho)

20. Critical value of z (one-tailed test for directional Ho)

21. Example - one sample z test
Calculated z = 19.35
Critical z = 1.65
(from z table,  = .05) (z.05 = 1.65)
Since the calculated z is > z.05 , we reject the null hypothesis. The law school admission rate is statistically significantly higher following the advising program.

22. one sample z test
Practice problems