Investigation 118 female senators suppose that an

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Investigation 1.18: Female Senators Suppose that an alien lands on Earth, notices that there are two different sexes of the human species, and sets out to estimate the proportion of humans who are female. Fortunately, the alien had a good statistics course on its home planet, so it knows to take a sample of human beings and produce a confidence interval. Suppose that the alien happened upon the members of the 2015 U.S. Senate as its sample of human beings, so it finds 20 women and 80 men in its sample. (a) Use this sample information to form a 95% confidence interval for the actual proportion of all humans who are female. (b) Is this confidence interval a reasonable estimate of the actual proportion of all humans who are female? (c) Explain why the confidence interval procedure fails to produce an accurate estimate of the population parameter in this situation. (d) It clearly does not make sense to use the confidence interval in (a) to estimate the proportion of women on Earth or even the U.S., but does the interval make sense for estimating the proportion of women in the 2015 U.S. Senate? Explain your answer. Discussion: x First, statistical tests and confidence intervals do not compensate for the problems of a biased sampling procedure. If the sample is collected from the population in a biased manner, the ensuing confidence interval will be a biased, and potentially misleading, estimate of the population parameter of interest.
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Chance/Rossman, 2015 ISCAM III Example 1.1 119 x A second important point to remember is that confidence intervals and significance tests use sample statistics to estimate population or process parameters. When the data at hand constitute the entire population of interest, then constructing a confidence interval from these data is meaningless. In this case, you know precisely that the proportion of women in the population of the 2015 U.S. Senators is 0.20 (exactly! no margin-of-error!), so it is senseless to construct a confidence interval from these data. Example 1.1: Predicting Elections from Faces? Try these questions yourself before you use the solutions following to check your answers. Do voters make judgments about a political candidate based on his/her facial appearance? Can you correctly predict the outcome of an election, more often than not, simply by choosing the candidate whose face is judged to be more competent-looking? Researchers investigated this question in a study published in Science (Todorov, Mandisodka, Goren, and Hall, 2005). Participants were shown pictures of two candidates and asked who has the more competent looking face. Researchers then predicted the winner to be the candidate whose face was judged to look more competent by most of the participants. For the 32 U.S. Senate races in 2004, this method predicted the winner correctly in 23 of them.
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