Alan, while snooping around his grandmother\'s basement stumbled upon a shiny ob

Question

Alan, while snooping around his grandmother's basement stumbled upon a shiny object protruding from under a stack of boxes. When he reached for the object a genie miraculously materialized and stated: "You have found my magic coin. If you flip this coin an infinite number of times you will notice that heads will show 60% of the time." Soon after the genie's declaration he vanished, never to be seen again. Alan, excited about his new magical discovery, approached his friend Ken and told him about what he had found. Ken was skeptical of his friend's story, however, he told Alan to flip the coin 100 times and to record how many flips resulted with heads. What is the probability that Alan will be able convince Ken that his coin has special powers by finding a p value below 0.05 (one tailed). Use the Binomial Calculator (and some trial and error) If Ken told Alan to flip the coin only 20 times, what is the probability that Alan will not be able to convince Ken (by failing to reject the null hypothesis at the 0.05 level)?

Explanation / Answer

a) p = 0.05 corresponds to z = 1.65

hypothetical mean of unbiased coin: 100(0.5) = 50
hypothetical variance of unbiased coin: 100(0.5)(0.5) = 25
hypothetical std of unbiased coin: sqrt(25) = 5

reject null hypothesis if you observe heads = 50 + 5*1.65 (or higher)

Now do similar for part b:

b)p = 0.05 corresponds to z = 1.65

hypothetical mean of unbiased coin: 20(0.5) = 10

hypothetical variance of unbiased coin: 20(0.5)(0.5) = 5
hypothetical std of unbiased coin: sqrt(5) = 2.236

reject null hypothesis if you observe heads = 10+ 2.236*1.65 (or higher)

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