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8.4 A Significance Test for the Difference of Two Proportions Reading Guide In the previous section, you learned how to estimate the _______ of the difference of any two _____________________. But sometimes you must ____________ between _____ alternatives. For example: Are ________________________________________________ more likely to be seriously injured? Have _____________________________________________ changed over the past 5 years? Is there a difference between the proportions of _____________________________________________________? The __________ of the difference involved is ___________________________. All you care about is whether you have enough __________ to conclude there is a ______________________. In this section, you will learn to perform a _______ ______________________________ in order to decide if the _________________________________ can reasonably be attributed to ___________________________ or if the _________________________________________ that something other than _____________________________ must be causing it. A Sampling Distribution of the Difference You will learn about what ________________________________ you should expect in your samples when there is ____ ___________________ in the proportions of success in the two ________________. Approximate sampling distributions for samples sizes ___, ___, and ___ are shown below. In each case the proportion of successes in the first population is ___, and the proportion of successes in the second population is ___. Note three facts about these approximate sampling distributions: Each sampling distribution is _____________________________ in shape and becomes more so with ___________ sample sizes. The mean of each sampling distribution is at ______________________________________. The formula to find the SE (_______________________) is the same as before: 1 The Theory of a Significance Test for the Difference of Two Proportions Example: Are Men More Likely to Be Left-Handed? Perform a ___________________ to determine whether males are more likely _________________________ than ________. 1. Check conditions: 2. Let Let p1 denote the proportion of __________________________________________________________________ p2 denote the proportion of __________________________________________________________________ The null hypothesis is always one of ______________________ or ______________________, so the appropriate null hypothesis is: The alternative hypothesis, sometimes called the ____________________, is a statement of what the researcher is _________________________________________. Here you are looking for evidence that the proportion of _____________ is greater ____________________. So the alternative hypothesis is: 3. Compute the test statistic and P-value. The __________________________ builds the best estimates of these _________ ________________________, namely the sample proportions: The general form of a ______________________________________is: The difference from the sample (__________________) is ___________________________. The hypothesized difference (______________________) is ___, the value under the null hypothesis ________________. The standard error of the estimate is given exactly by: You could estimate ___________ and ______________. However, you can do even better. The null hypothesis states that the ____________________________________ is equal to ___________________________________, that is __________. You can estimate this common value of _________ by combining data from other samples into a _____________________, ___. The pooled estimate is found by combining males and females into one group: 2 The test statistic is equal to: _________, with a P-value of: __________. The P-value is very __________, so you _________ the null hypothesis. The difference between the rates of ______________ in these two samples is too large to attribute to _______________________. The evidence supports the alternative that _______________________________________________________. 3 4