Go to the Reese\'s pieces samples applet http: //www.rossmanchance.com/applets/O

Question

Go to the Reese's pieces samples applet http: //www.rossmanchance.com/applets/OneProp/OneProp.htm?candy=1 To run it you need to click on the handle on the drawn candy dispenser, as if you were buying the candy from the machine. Get the distribution of cap p (the proportion of orange candy dispensed) for sample size of n = 50 and pi = 0.03. To get the distribution, you are simulating the CLT empirically. You are doing a simulation. Each trial of the simulation entails drawing a sample of size 50. You conduct 1000 trials. For each sample of 50 (i.e., for each trial), the applet is computing the cap p the proportion of the 50 candies that are orange. So you get 1000 cap ps. These 1000 phats are then counted to see how many are less than 0.01, how many between 0.01 and 0.02, etc... The histogram shown summarizes how often you get a given value of a particular phat. (a) Describe the distribution that you get and comment on the following: Is the normal a good approximation to the distribution of cap p in this case? Why? Refer to your the distribution you get. (b) What can you control and would have to change in the simulation to have a distribution of cap p that is normal? Explain why. Refer to the theory learned in class.

Explanation / Answer

a. The right tail of the distribution is far extended than the left tail. Therefore, the distribution is skewed. Check for the success/failure condition. One expects, np=50*0.03=1.5 and n(1-p)=50*0.97=48.5. Both not far greater than 10. Thus, normal is not a good approximation to the distribution of phat.

b. In order to have a distribution of phat that is normal, increase the sample size so that there exists atleast 10 expected success and 10 expected failures.

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