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# Aa calculate and interpret a 95 confidence interval

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(aa) Calculate and interpret a 95% confidence interval to compare hearing loss of American teenagers in these two years. Is this confidence interval consistent with your test of significance? Explain.

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Chance/Rossman, 2015 ISCAM III Investigation 3.1 193 Note: It is technically incorrect to say there has been a 0.1% to 4.7% increase, because “percentage change” implies a multiplication of values, not an addition or subtraction as we are considering here. It would be acceptable to say that the increase is between 0.1 and 4.7 percentage points . Technical Conditions The above Central Limit Theorem holds when the populations are much larger than the samples (e.g., more than 20 times the sample size) and when the sample size is large. We will consider the latter condition met when we have at least 5 successes and at least 5 failures in each sample (so there are four numbers to check). Note: A “Wilson adjustment” can be used with this confidence interval as before, this time putting one additional success and one additional failure in each sample. This adjustment will be most useful when the proportions are close to 0 or 1 (that is when the sample size conditions above are not met). (bb) Summarize your conclusions from this study. Be sure to address statistical significance, statistical confidence, and the populations you are willing to generalize the results to. Also, are you willing to conclude that the change in the prevalence of hearing loss is due to the increased use of ear buds among teenagers between 1994 and 2006? Explain why or why not. Study Conclusions We have moderate evidence against the null hypothesis (p-value | 0.02, meaning we would get a difference in sample proportions 1 ˆ p ± 2 ˆ p as small as ± 0.024 or smaller in about 2% of random samples from two populations with S 1 = S 2 ). We are 95% confident that the population proportion with some hearing loss is between 0.001 and 0. 047 higher “now” than ten years ago. We feel comfortable drawing these conclusions about the populations the NHANES samples were selected from as they were random samples from each population (and there was no overlap in the populations between these two time periods). However, there are many things that have changed during this time period, and it would not be reasonable to attribute this increase in hearing loss exclusively to the use of ear buds. Practice Problem 3.1A (a) When we conducted the simulation analysis above, we used the same probability of “success” (having some hearing loss) for both years. Why did we do this? (b) How did we decide what common probability of success to use? (c) Why did we count how many samples of the simulation gave a result of ± 0.024 or smaller (explain the ± 0.024 part and the “or smaller” par t).
Chance/Rossman, 2015 ISCAM III Investigation 3.1 194 Technology Detour Two-sample z-procedures Theory-Based Inference applet x Select Two proportions x Check the box to paste in 2 columns of data (stacked or unstacked) and press Use data or specify the sample sizes and either the sample counts or the sample

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