(A) Figure 56.1 is a histogram showing the results of 1000 repetitions of flipping a fair coin 100 times. The results are the number of times the 100 flips came up with various proportions of heads. Does this look approximately like a normal distribution? Eyeballing this histogram, what is the approximate 95% confidence interval for a fair coin? What is the approximate 99% confidence interval?
Answer: Yes, it approximates a normal distribution. It looks like about 950 of the 1000 proportions (95%) are in the interval 0.4–0.6. About 50 (5%) of the results have proportions outside this interval, with about 25 (2.5%) on each side (in each tail). Also, although it is hard to eyeball, about 990 of the 1000 proportions (99%) are betwen about 0.37 and about 0.63 inclusive, with about 5 (0.5%) in each tail outside this interval. (Simulating 10,000 repetitions would be better.)
(B) Looking at Table 56.1, check whether your eyeballed 95% confidence interval matches the theoretical results one should obtain when flipping fair coins 100 times. (Recall that the margin of error is the half-width of the 95% confidence interval and can be expressed as a percentage rather than a proportion.)
|Margin of error (%)||±9.80||±4.38||±3.10||±2.53|
Answer: Yes, it does. 50% ± 9.8% is about 40–60%, or 0.4–0.6 proportions.