Notice that the only difference between these two

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Notice that the only difference between these two formulas is the first factor: 1 ± ± N n N . This is referred to as a “finite population correction factor.” It tells us the factor by which we should decrease the measure of sample to sample variability in the statistic. This multiplier does approach one as N approaches infinity, or in other words, as our population becomes much, much larger than our sample. For the Gettysburg example with n = 40, this would violate our “large population” technical condition. So we would calculate 40 ) 0466 . 1 ( 466 . 0 1 268 40 268 ± ± ± = 0.0729 instead of ± 40 ) 0466 . 1 ( 466 . 0 0.0799. This standard deviation is a more accurate measure of the sample to sample variability when the population size is not very large.
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Chance/Rossman, 2015 ISCAM III Investigation 1.16 114 Investigation 1.16: Teen Hearing Loss Shargorodsky, Curhan, Curhan, and Eavey (2010) examined hearing loss data for 1771 participants from the National Health and Nutrition Examination Survey (NHANES, 2005-6), aged 12-19 years. ( “NHANES provides nationally representative cross -sectional data on the health status of the civilian, non-institutionalized U.S. population.” ) In this sample, 333 were found to have at least some level of hearing loss. News of this study spread quickly, many blaming the prevalence of hearing loss on the higher use of ear buds by teens. At MSNBC.com (8/17/2010), Carla Johnson summarized the study with the headline “1 in 5 U.S. teens has hearing loss, study says.” We will first decide whether we consider this an appropriate headline. (a) Define the population and parameter of interest in this study (in symbols and in words). Population: Parameter: (b) State the null and alternative hypotheses for deciding whether these data provide convincing evidence against this headline (in symbols and in words). Discussion: A consequence of the expansion of the Central Limit Theorem we made in the previous investigation is that we can use all the same normal-based z -procedures when we have a large enough sample size and have taken a random sample from a large population. (c) Under the null hypothesis, is the normal model appropriate for the distribution of sample proportions based on this sample size and on this population size? Justify your answers numerically. (d) Use technology to carry out a z -test of the hypotheses you specified in (b). Report the test statistic and p-value, and confirm how you would calculate the test statistic by hand.
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Chance/Rossman, 2015 ISCAM III Investigation 1.16 115 (e) Does this p-value provide convincing evidence ( D = 0.05) against the headline? Justify your answer (in context) and explain the reasoning behind your evidence or lack thereof. (f) Calculate and interpret a 95% confidence interval. Is this interval consistent with your test decision? (g) Do you feel comfortable with generalizing the findings from your test and confidence interval to the population of all American teens in 2005-06? Explain.
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