invest_3ed.pdf

# Shape mean standard deviation comparison sample means

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Shape: Mean: Standard deviation: Comparison (sample means vs. population): (m) How can we use your simulated distribution of sample means to decide whether it is surprising that a boat with 47 passengers would exceed the (average) weight limit by chance (random sampling error) alone? x

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Chance/Rossman, 2015 ISCAM III Investigation 2.4 153 (n) To investigate the question posed in (m), specify the sample mean of interest (159.574) in the Count Sample box, use the pull-down menu to specify whether you want to count samples Greater Than , Less Than , or Beyond (both directions). Press Count . What conclusions can you draw from this count? Non-normal Population To carry out the preceding simulation analysis, we assumed that the population distribution had a normal shape. But what if the population of adult weights has a different, non-normal distribution? Will that change our findings? (o ) Now copy and paste the data from the “ pop2 ” column in the WeightPopulations.xls data file. Describe the shape of this population and what it means for the variable to have this shape in this context. How do the values of P and V compare to the previous population? Shape: Mean: Standard deviation: Comparison: (p) Generate 1000 random samples of 47 individuals from this population. Produce a well-labeled sketch of the distribution of sample means below and note the values for the mean and standard deviation. How does this distribution compare to the one in (l)? Sketch: Mean: SD: Comparison: (q) Again use the applet to approximate the probability of obtaining a sample mean weight of at least 159.574 lbs for a random sample of 47 passengers from this population. Has this probability changed considerably with the change in the shape of the distribution of the population of weights? (r) Repeat this analysis using the other population distribution ( pop3 ) in the data file and summarize your observations for the three populations in the table below. Distribution of sample means Shape Mean Standard deviation Normal ( P = 167, V = 35) Skewed right ( P = 167, V = 35) Uniform ( a = 106.4, b = 227.6)
Chance/Rossman, 2015 ISCAM III Investigation 2.4 154 (s) Now consider changing the sample size from 47 to 188 (four times larger). Make a prediction for how the shape, mean, and SD of the distribution of sample means would change (if at all). (t) Make this change of sample size from 47 to 188, and generate 1000 random samples (from the uniformly distributed population, pop3). Did the shape of the sample means change very much? What about the mean of the sample means? What about the SD of the sample means? How were your predictions in (s)? Discussion: You should see that, as with the Gettysburg Address investigation, the shape of the population is not having much effect on the distribution of the sample means! In fact, you can show that the mean of the distribution of sample means from random samples is always equal to the mean of the population (any discrepancies you find are from not simulating enough random samples) and that, assuming the population is large compared to the size of the sample, the standard deviation of the

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