invest_3ed.pdf

# G based on this calculation would you consider the

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(g) Based on this calculation, would you consider the value of the sample mean (98.249) to be surprising, if the population mean were really equal to 98.6? Explain how you are deciding. Previously, we compared our standardized statistics ( z -scores) to the normal distribution and said (absolute) values larger than two were considered rare. Is that still true when we have used the sample standard deviation in our calculation? L et’s explore the method you just used to standardize the sample mean (using the standard error) in more detail. (h) Open the Sampling from a Finite Population applet and paste the hypothetical population body temperature data from the BodyTempPop.txt file. Does this appear to be a normally distributed population? What are the values of the population mean and the population standard deviation? (i) Use the applet to select 10,000 samples of 13 adults from this hypothetical population. Confirm that the behavior of the distribution of sample means is consistent with the Central Limit Theorem? [ Hint : Discuss shape, center, and variability; compare predicted to simulated.] (j) Where does the observed sample mean of 98.249 fall in this sampling distribution? Does it appear to be a surprising value if the population mean equals 98.6?

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Chance/Rossman, 2015 ISCAM III Investigation 2.5 159 So if we have a value for V , we are fine. But what about the statistic suggested in (f); does this standardized statistic behave nicely and is this distribution again well modeled by a normal distribution? (k) In the applet, change the Statistic option (above the graph) to t -statistic , the name for the standardized sample mean using the standard error of the sample mean. Describe the shape of the distribution of these standardized statistics from your 10,000 random samples. (l) Check the box to Overlay Normal Distribution ; does this appear to be a reasonable fit? What p- value does this normal approximation produce? [ Hint : Enter your answer to (f) as the observed result for the t -statistic and count beyond.] (m) Does the theory-based p-value from the normal distribution accurately predict how often we would simulated a standardized statistic at least as extreme (in either direction) as the observed value of ± 1.73? Does it over predict or underpredict? [ Hint : How does the behavior of the distribution of the standardized statistics most differ from a normal model?] Discussion: If we zoom in on the tails of the distribution, we see that more of the simulated distribution lies in those tails than the normal distribution would predict. To model the sampling distribution of the standardized statistic ( x ± P )/( s / n ), we need a density curve with heavier tails than the standard normal distribution. William S. Gosset, a chemist turned statistician, showed in 1908, while working for the Guinness Breweries in Dublin, that a t probability curve (see next page) provides a better model for the sampling distribution of this standardized statistic when the population of observations follows a normal distribution.
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