1. STA 291 Summer 2008
Lecture 18
Dustin Lueker

2. Significance Test
A way of statistically testing a hypothesis by comparing the data to values predicted by the hypothesis
Data that fall far from the predicted values provide evidence against the hypothesis
STA 291 Summer 2008 Lecture 18

3. Logical Procedure
State a hypothesis that you would like to find evidence against
Get data and calculate a statistic
Sample mean
Sample proportion
Hypothesis determines the sampling distribution of our statistic
If the calculated value in 2 is very unreasonable given 3, then we conclude that the hypothesis is wrong
STA 291 Summer 2008 Lecture 18

4. Elements of a Significance Test
Assumptions
Type of data, population distribution, sample size
Hypotheses
Null hypothesis H0
Alternative hypothesis H1
Test Statistic
Compares point estimate to parameter value under the null hypothesis
P-value
Uses the sampling distribution to quantify evidence against null hypothesis
Small p-value is more contradictory
Conclusion
Report p-value
Make formal rejection decision (optional)
Useful for those that are not familiar with hypothesis testing
STA 291 Summer 2008 Lecture 18

5. P-value
How unusual is the observed test statistic when the null hypothesis is assumed true?
The p-value is the probability, assuming that the null hypothesis is true, that the test statistic takes values at least as contradictory to the null hypothesis as the value actually observed
The smaller the p-value, the more strongly the data contradicts the null hypothesis
STA 291 Summer 2008 Lecture 18

6. Conclusion
In addition to reporting the p-value, sometimes a formal decision is made about rejecting or not rejecting the null hypothesis
Most studies require small p-values like p<.05 or p<.01 as significant evidence against the null hypothesis
“The results are significant at the 5% level”
α=.05
STA 291 Summer 2008 Lecture 18

7. Example
Which p-value would indicate the most significant evidence against the null hypothesis?
.98
.001
1.5
-.2
STA 291 Summer 2008 Lecture 18

8. Rejection Region
Range of values such that if the test statistic falls into that range, we decide to reject the null hypothesis in favor of the alternative hypothesis
Type of test determines which tail(s) the rejection region is in
Left-tailed
Right-tailed
Two-tailed
STA 291 Summer 2008 Lecture 18

9. Examples
Find the rejection region for each set of hypotheses and levels of significance.
H1: μ > μ0
α = .05
H1: μ < μ0
α = .02
H1: μ ≠ μ0
α = .01
STA 291 Summer 2008 Lecture 18

10. Test Statistic The z-score has a standard normal distribution
The z-score measures how many estimated standard errors the sample mean falls from the hypothesized population mean
The farther the sample mean falls from the larger the absolute value of the z test statistic, and the stronger the evidence against the null hypothesis
STA 291 Summer 2008 Lecture 18

11. Example
Thirty-second commercials cost $2.3 million during the 2001 Super Bowl. A random sample of 116 people who watched the game were asked how many commercials they watches in their entirety. The sample had a mean of and a standard deviation of Can we conclude that the mean number of commercials watched is greater than 15?
State the hypotheses, find the test statistic and rejection region for testing whether or not the mean has changed, interpret
Make a decision, using a significance level of 5%
STA 291 Summer 2008 Lecture 18