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STA 291 Fall 2009

STA 291 Fall 2009. Lecture 19 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. Logical Procedure.

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STA 291 Fall 2009

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  1. STA 291Fall 2009 Lecture 19 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 Fall 2009 Lecture 19

  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 Fall 2009 Lecture 19

  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 Fall 2009 Lecture 19

  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 Fall 2009 Lecture 19

  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 Fall 2009 Lecture 19

  7. Example • Which p-value would indicate the most significant evidence against the null hypothesis? • .98 • .001 • 1.5 • -.2 STA 291 Fall 2009 Lecture 19

  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 Fall 2009 Lecture 19

  9. Examples • Find the rejection region for each set of hypotheses and levels of significance. • H1: p > p0 • α = .05 • H1: p < p0 • α = .02 • H1: p ≠ p0 • α = .01 STA 291 Fall 2009 Lecture 19

  10. Test Statistic • The z-score has a standard normal distribution • The z-score measures how many estimated standard errors the sample proportion falls from the hypothesized population proportion • The farther the sample proportion falls from p0 the larger the absolute value of the z test statistic, and the stronger the evidence against the null hypothesis STA 291 Fall 2009 Lecture 19

  11. Example • In the 2006 Major League Baseball season there were 2429 regular season games. It turns out that the home team won 1327 of the 2429 games, or 54.63% of the time. Test at a 5% level of significance whether or not there is a home field advantage in MLB or if the percentage being different from 50% is due to mere chance. Use the rejection region method. STA 291 Fall 2009 Lecture 19

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