Chapter 20: Testing Hypotheses about Proportions “Half the money I spend on advertising is wasted; the trouble is I don’t know which half.” -John Wanamaker

Hypotheses Hypothesis: a supposition a proposition or principle which is supposed or taken for granted, in order to draw a conclusion or inference for proof of the point in question In statistical tests of hypotheses, we assume that a hypothesis is true. If the data are consistent with the hypothesis, we retain the hypothesis If the data are not consistent with the hypothesis, we reject the hypothesis

Testing Hypotheses The Null Hypothesis The starting hypothesis The null hypothesis specifies a population model parameter of interest and proposes a value for the parameter Standard Deviation (not Standard Error) We assume the null hypothesis to be true, so we have a value for the model parameter p.

The Reasoning of Hypothesis Testing Hypotheses The null hypothesis Translate our question into a statement about model parameters Write The alternative hypothesis Contains the values of the parameter we accept if we reject the null.

The Reasoning of Hypothesis Testing Plan Specify the model you will use to test the null hypothesis and the parameter of interest All models require assumptions, so state them and check any corresponding conditions Include the name of the test you plan to perform End with a statement such as “Because the conditions are satisfied, it id appropriate to model the sampling distribution of the proportion with a model”

The Reasoning of Hypothesis Testing Mechanics Do the actual calculations of a test statistic from the data The ultimate goal is to obtain a P-value The probability that the observed statistic value could occur if the null model were correct If the P-value is small enough, reject the null hypothesis

The Reasoning of Hypothesis Testing Conclusion The conclusion in a hypothesis test is always a statement about the null hypothesis “reject” or “fail to reject” the null hypothesis Consider the size of the effect by examining a confidence interval

One-proportion z-test The conditions for the one-proportion z-test are the same as for the one-proportion z-interval. When the conditions are met and the null hypothesis is true, the statistic follows the standard Normal model, so we can use that model to obtain a P-value.

Alternative Alternatives Two-sided alternative When we are equally interested in proportions that deviate from p in either direction

Alternative Alternatives One-sided alternative An alternative hypothesis that focuses on deviations from the null hypothesis value in only one direction. The P-value is the probability of deviating only in the direction of the alternative away from the null hypothesis value

TI-83+ Tips STAT TESTS 5: 1-Prop ZTest

TI-83+ Tips Specify the hypothesized proportion Po Enter x, the observed number of wins Specify the sample size Decide on one- or two-tailed test Calculate

P-Values and Decisions: What to Tell About a Hypothesis Test Hypothesis tests are useful when making a decision It is a good idea to report the confidence interval for the parameter of interest The P-value is highly context-dependent The importance of the issue also factors into the choice of P-value The conclusion about any null hypothesis should be accompanied by the P-value of the test. If possible, it should also include a confidence interval for the parameter of interest