STA 291 Spring 2008 Lecture 18 Dustin Lueker

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 2008 Lecture 18

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 2008 Lecture 18

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 2008 Lecture 18

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 2008 Lecture 18

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 2008 Lecture 18

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

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 2008 Lecture 18

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 2008 Lecture 18

Test Statistic Testing µ with σ unknown and n large Just like finding a confidence interval for µ with σ unknown and n large Reasons for choosing test statistics are the same as choosing the correct confidence interval formula STA 291 Spring 2008 Lecture 18

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 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 rejection region for testing whether or not the mean has changed, interpret Make a decision, using a significance level of 5% STA 291 Spring 2008 Lecture 18