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STA 2023 Honors Fall 2007 Name: _______________________ Project 3 Purpose: To study the sampling distribution of the sample mean and the sample proportion using Java Applets on the Internet. Note – All the applets for this project can be found at http://www.stat.ufl.edu/~mripol/Applets/SamplingDistributionApplets.html Agresti Applets better now. 1. Two fun applets: a) Falling balls at http://www.rand.org/methodology/stat/applets/clt.html This applet illustrate how a Normal distribution arises naturally from random variability. The Central Limit Theorem says that the distribution of averages is approximately Normal. Many things in nature (heights, weights, diameters, etc) are approximately Normal because they are the result of many small random effects. A person’s height, for example, is determined by a lot of different factors – genes from mother and father, food intake through childhood, childhood illnesses, exercise, etc. b) Rolling dice at http://www.stat.sc.edu/~west/javahtml/CLT.html Rolling one die gives you a Uniform Distribution. You can increase the number of rolls to do this faster. Rolling several dice and recording the sum gives you a distribution that approaches Normality as n increases. 2. Studying the Normal Approximation to the Binomial. In class, we studied the sampling distribution of p-hat, the sample proportion of successes in a Binomial experiment. We saw that this distribution is approximately Normal if np and n(1-p) are both greater than or equal to 15. The same conditions are necessary for the distribution of X=number of successes in a Binomial experiment to be approximately Normal. We will study this with the applet at http://bcs.whfreeman.com/ips5e. Click on Statistical Applets, then Normal Approximation to the Binomial. a) b) Fill out the table below: For each setting of n and p given, compute the values of np and n(1-p). Then use the applet to see the graph and determine if the Normal approximation is good for each case. Does graph show Binomial close to Normal? Look at symmetry, continuity and tails. Careful with the value of p on the last three rows!! n p np n(1-p) both ≥ 15? Normal approximation good? If not, why? 20 0.70 30 0.70 80 0.70 10 0.50 15 0.50 40 0.50 10 0.03 50 0.03 100 0.03 Play with the applet a bit. In your own words, explain what combinations of n and p give a Binomial Distribution that is approximately Normal. 3. Studying the Central Limit Theorem: X ~ N , for n large enough. n Use the applet at http://www.ruf.rice.edu/%7Elane/stat_sim/sampling_dist/index.html. Starting with parent populations of different shapes (selected from the drop-down menu), we will increase n to see how the distribution of X changes. Make sure that for the bottom two graphs, the Mean is selected on the drop-down menu. a) b) Fill out the table below: For each parent distribution and sample size given, compute the values mean and standard deviation of the distribution of X . Then use the applet to get the distribution (get at least 10,000 samples). Record the mean and standard deviation of your simulation. Comment on the shape of the graph. For the “custom” parent population, use the mouse to create a Bimodal distribution. NOTE – the resolution of the graphs is not very good. When you look at the shape, imagine it being smoother. Parent Population n Normal μ= 16 σ=5 Normal μ= 16 σ=5 Uniform μ= 16 σ=9.52 Uniform μ= 16 σ=9.52 Skewed μ= 8.08 σ=6.22 Skewed μ= 8.08 σ=6.22 Custom-Bimodal μ= σ= Custom-Bimodal μ= σ= 2 Theoretical Mean Stdev Observed Mean Stdev Shape of Sampling Distn of X 25 2 25 2 25 2 25 In your own words, explain when the distribution of the sample mean will be approximately Normal.