STA 291 Spring 2010

1. STA 291Spring 2010 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 Spring 2010 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 Spring 2010 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 Spring 2010 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 Spring 2010 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 Spring 2010 Lecture 18

7. Example • Which p-value would indicate the most significant evidence against the null hypothesis? • .98 • .001 • 1.5 • -.2 STA 291 Spring 2010 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 Spring 2010 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 Spring 2010 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 Spring 2010 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 watched in their entirety. The sample had a mean of 15.27 and a standard deviation of 5.72. Can we conclude that the mean number of commercials watched is greater than 15? • State the hypotheses, find the test statistic and use the rejection region • Make a decision, using a significance level of 5% STA 291 Spring 2010 Lecture 18