Suppose that you hold a coin vertically on edge, flick one side to make it spin, then see if it settles on heads or tails. Is this a fair process? That is, will half of spins tend to fall heads and the other half on tails? To investigate this question, a student spun the same US penny 200 times and recorded 84 heads and 116 tails. She uses these results to test H₀: p = 0.5 vs H_a: p ≠ 0.5, where p is the proportion of penny spins that land heads. The file RandomP50N200 contains counts and proportions of heads for 5000 simulated samples of size n = 200 when p = 0.50.
(a) Use the randomization proportions, the variable Phat in RandomP50N200, to estimate the standard error for the sample proportion, p̂, in this situation.
(b) Use the standard error from part (a) to compute a standardized test statistic for testing H₀: p = 0.5 based on her original sample of 200 spins.
(c) Under this null hypothesis, what should be the mean of the distribution of counts of number of heads in 200 spins?
(d) The counts of heads for each of the 5000 randomization samples are in the variable Count. Use these to estimate the standard deviation for the counts (number of heads in 200 spins) under this null hypothesis.
(e) Use the mean from part (c) and standard error from part (d) to compute a standardized test statistic based on the count of 84 heads in the original sample of 200 spins. How does this test statistic compare to the one based on the sample proportions in part (b)?
(f) Use either test statistic and the fact that both randomization distributions are relatively normal to compute and interpret a p-value for this test.