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Unformatted text preview: Lecture 18 Nancy Pfenning Stats 1000 Chapter 9: Means and Proportions as Random Variables Recall: in Section 1.3, we stated that the normal curve is idealized: its curve depicts an idealized histogram from a population with infinite possible values, all falling into a precise pattern. We call its mean μ and its standard deviation σ . Example Height (a quantitative variable) of college-aged women in the U.S. is normal with μ = 65 ,σ = 2 . 7. Of course this is idealized—we haven’t measured all of them! On the other hand, a set of actual observations from a variable x has a mean ¯ x and a standard deviation s . Example Heights of women in this class have mean ¯ x = . 75 in , standard deviation s = . 75 in . Analogously for categorical variables, Example The proportion of women in the U.S. is p = . 5. Example The proportion of women in this class is ˆ p = 1 in . In the first and third examples, μ and σ , p are parameters , numbers which describe the entire population . In the second and fourth examples, ¯ x and s , ˆ p are statistics , numbers computed or measured from sample data. Example Identify the sample, the population, the statistic, and the parameter of interest in each of the following: 1. A survey is carried out at a university to estimate the proportion of undergraduates living at home during the current term. sample : the undergrads surveyed; population : all undergrads at that university; statistic : proportion of sampled undergrads living at home; parameter : proportion of all undergrads at that university living at home. 2. In 1988, investigators chose 400 teachers at random from the National Science Teachers Association list and polled them as to whether or not they believed in the biblical creation. Of 200 respondents, 30% did believe. sample : 200 respondents; population : National Science Teachers Association members; statistic : 30% (proportion of believers in sample); parameter : unknown proportion of all NSTA members believing in biblical creation. 3. A survey of 1000 households in a certain city found their mean household size to be ap- proximately 3.1 persons. sample : 1000 households surveyed; population : all households in that city; statistic : 3.1 (mean size of sampled households); parameter : unknown mean household size in city. 4. A balanced coin is flipped 100 times and the percentage of heads is 47%. sample : the 100 flips; population : all coin flips; statistic : 47%; parameter : 50% (percentage of all coinflips that would result in heads). 72 Ultimately, we will measure statistics and use them to draw conclusions about unknown parameters [statis- tical inference, or reasoning “backward”]. First, we must discover, for a given parameter, how the accompanying statistic tends to behave [reasoning “forward”, which is accomplished through use of the laws of probability]. This forward reasoning process is in a way impractical, because in real life parameters are usually unknown and cannot be “given”. But wein a way impractical, because in real life parameters are usually unknown and cannot be “given”....
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This note was uploaded on 02/15/2012 for the course STAT 1000 taught by Professor Taeyoungpark during the Fall '06 term at Pittsburgh.
- Fall '06