Winter, 2012 Thursday, Feb. 2Stat 217 – Day 17Two-sample z-proceduresRecap:- Want to compare two proportions from independent random samples or randomized experiment with one procedureo With random sampling, we can generalize our results (significant or not) to the largerpopulation(s).o With a randomized, comparative experiment have the potential to draw a cause-and-effect conclusion between the explanatory and response variable A double blind study, which may involve the use of a placebo treatment, is another way to guard against any other confounding variables affecting the treatment groups differently (e.g., power of suggestion)o H0: 1 – 2 = 0 (no difference in population proportions or treatment probabilities)- When the sample sizes are large, the distribution of the difference between two sample proportions can be approximated by a normal distribution with mean equal to 1 – 2 and standard deviation 222111/)1(/)1( nn. o p-value is the percentage of random samples/random assignments that produce values of pˆ1- pˆ2 at least as extreme as those observed in the research study.o The sample sizes will be considered large enough as long as there are at least 5 successes in each group and at least 5 failures in each sample.Investigation 4: Chocolate Lovers Live Longer (to be submitted with a partner by Tuesday)Newspaper headlines proclaimed that chocolate lovers may live longer, following the publication of a study title “Life is Sweet: Candy Consumption and Longevity” in the British Medical Journal (Lee and Paffenbarger, 1998). Researchers speculate that antioxidants present in chocolate may have a health benefit. In 1988, researchers sent a health questionnaire to men (free of cardiovascular disease and cancer) who entered Harvard University as undergraduates between 1916 and 1950, asking about their candy habits over the past year. From the questionnaire they determined whether the respondents ate a “moderate” amount of sweets (allowing themselves only one to three candy bars a month) or were non-candy eaters (“almost never”). Then they tracked the participants to determined whether or not they had died by 1993. Here are the data:Moderate candy consumer Nonconsumer TotalHad died 267 247 514Had not died 4262 3065 7327Total 4529 3312 7841(a) Identify the observational units in this study. (b) Identify the two variables in this study.Winter, 2012 Thursday, Feb. 2(c) Which variable would you consider the explanatory variable and which the response variable? (d) Is this an observational study or an experiment? Justify your answer.(e) Do these data provide preliminary evidence of an association between candy consumption and life expectancy? Support your answer with appropriate numerical calculations.(e) Consider these samples to be representative of adult American males. Define the parameter of interest and state the null and alternative hypotheses, in symbols and in words, for testing whether the probability of a death is lower for candy consumers.Parameter:Null hypothesis, H0:Alternative hypothesis, Ha:(f) Use the Two-way Table Inference applet to approximate the p-value for this test. (g) Are the sample size conditions necessary for the Central Limit Theorem to be valid met for this study? Justify your answer.(h) Use the Test of Significance Calculator applet to approximate the p-value for this test.- Use the pull-down menu to select Two proportions- Specify zero as the hypothesized difference- Click on the < (if necessary) to toggle the inequality to match your Ha- Enter the sample sizes and number of deaths for each group (as you tab over,confirm the calculations of the conditional proportions).- Press Calculate.The applet will shade the area representing the p-value and display the result in the p-value box.Winter, 2012 Thursday, Feb. 2(i) Report and interpret this p-value.When the p-value is small, it is often useful to also have a measure of the distance between the observed statistic and the hypothesized value for the parameter. But we will always measure distances in terms of the number of standard deviations between the two values. Wecould calculate this value using the standard deviation formula specified by the Central Limit Theorem.22211121)1()1(0ˆˆnnppz[Note: To actually calculate this value, we have to put in estimates for 1 and 2, but don’t worry about those details right now.] In general, this calculation is called the test statistic and provides a measure of the discrepancy between what we observed in the research study and the null hypothesis. With proportions, this is called a z-statistic and based on the Empirical Rule, we know values more than 2 or 3 are pretty surprising. [If you p-value is less than .001, then it’s useful to know whether the z value is 4 or 40!] The procedure you have just applied is often called a two-sample z-test.(j) Report (from the applet output) and interpret the z-value for this study.(k) Based on your p-value and test statistic, would you reject or fail to reject the null hypothesisin favor of the alternative hypothesis?Rather than only answering whether or not we have convincing evidence that there is a difference in the populations, it is often more useful to estimate the size of the difference. We do this with a confidence interval for 1 – 2.(l) Keep the confidence level set to 95% and press the Confidence Interval button. Report the confidence interval that displays on the button. This interval is often called a two-sample z-interval.Winter, 2012 Thursday, Feb. 2(m) Remember that this a confidence interval for the difference in the population proportions. Is zero contained in your confidence interval? Explain how you could have predicted this based on the results of your test of significance (your p-value). (n) Remember that this is a confidence interval for the difference in the population proportions. What is the implication of both values of the confidence interval being negative?(o) Find the midpoint of this confidence interval (average the two endpoints). Does this number look familiar?(p) Determine the margin-of-error for this confidence interval by taking the larger endpoint minus the smaller endpoint and dividing by two, the half-width of the interval. So the general form of this confidence interval is exactly as we saw beforeestimate + margin-of-errorwhere the estimate is equal to our observed statistic and the