V use the theoretical results for the mean and

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(v) Use the theoretical results for the mean and standard deviation of a sample mean to standardize the value of 159.57 lbs. [ Hint : Start with a well-labeled sketch.] How might you interpret this value? (w) Use the result from the Central Limit Theorem and technology (e.g, the Normal Probability Calculator applet) to estimate the probability of a sample mean weight exceeding 159.57 lbs for a random sample of 47 passengers from a population with mean P = 167 lbs and standard deviation V = 35 lbs. [ Hint : Shade the area of interest in your sketch in (u).] How does this estimated probability compare to what you found with repeated sampling from the hypothetical populations? (x) Identify one concern you might have with this analysis. [ Hint : What other assumption, apart from the shape of the population, was made in these simulations that may not be true in this study? Do you think this is a reasonable assumption for this study?]
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Chance/Rossman, 2015 ISCAM III Investigation 2.4 156 Study Conclusions Assuming the CDC values for the mean and standard deviation of adult Americans’ weights, μ = 167 lbs and σ = 35 lbs, we believe that the distribution of sample mean weights will be well modeled by a normal distribution (based both on the not extremely skewed nature of the variable and the moderately large sample size of 47, which is larger than 30). Therefore, the Central Limit Theorem allows us to predict that the distribution of x is approximately normal with mean 167 lbs and standard deviation 35/ 47 ≈ 5.105 lbs. From this information , assuming the CDC data is representative of the population of Ethan Allen travelers, we can estimate the probability of obtaining a sample mean of 159.57 pounds or higher to be 0.9264. Therefore, it is not at all surprising that a boat carrying 47 American adults capsized. In fact, the surprising part might be that it didn’t happen sooner! Practice Problem 2.4A (a) Use the Sampling from Finite Population applet or the Central Limit Theorem to estimate the probability that the sample mean of 20 randomly selected passengers exceeds 159.57lbs, assuming a normal population with mean 167lbs and standard deviation 35lbs. (b) Is the probability you found in (a) larger or smaller than the probability you found for 47 passengers? Explain why your answer makes intuitive sense. (c) Repeat (a) assuming a uniformly distributed population of weights. How do these two probabilities compare? [ Hint : Think about whether it is more appropriate to use the Sampling from Finite Population applet or the CLT to answer this question.] (d) Explain why the calculation in (a) does not estimate the probability of the Ethan Allen sinking with 20 passengers. Practice Problem 2.4B Use the Sampling from Finite Population applet or the Central Limit Theorem to estimate the probability that the sample mean of 47 randomly selected passengers would exceed 159.57lbs, assuming that random samples are repeatedly selected from a population of 80,000 individuals with mean 167 lbs and standard deviation 35 lbs. State any assumptions you need to make, and support your answer statistically.
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Chance/Rossman, 2015
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