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# Normal approximation to the binomial distribution

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A. Tossing a Coin

Simulate 6 independent tosses of a fair coin. Repeat the procedure to get 200 sets of 6 coin tosses. (That's 1200 total tosses.) List the results.

Explain carefully your method of simulating the tosses, whether it involves a computer, a table of random digits, or even flipping several coins at once. Address the issues of fairness, independence, and randomness. You are encouraged to find a clever way to generate the data.
Let X denote the number of heads in a set of 6 tosses. Thus the sample space for X is {0, 1, 2, 3, 4, 5, 6}. From your data above, make a histogram of the observed values of X. There are 200 cases.
From the data, compute the relative frequency of cases with X > 2.
In theory, what is the probability that X > 2 in any given case? Remember, we assume that the chance of heads on any toss is p = .5 and that the toss outcomes are independent.

Do you find that this agrees with the data, within reason?
Suppose just for now that p = .525. For this problem first compute the total number of heads in all 1,200 tosses. (Use your histogram.)

Next let n = 1200 and use the normal approximation to the binomial distribution B(1200, .525) to find the probability of observing no more than the number of heads you did.