Question

A South African mathematician, John Kerrich, was visiting Copenhagen in 1940 when Germany invaded Denmark. Kerrich was forced to spend the next five years in an internment camp, and to pass the time, he carried out a series of experiments. One such experiment involved flipping a coin 10,000 times and keeping track how many heads he obtained. Of all the 10,000 coin flips, 5067 came up heads.

a.Use the normal approximation to calculate a 95% confidence interval for the true probability of heads for Kerrich’s coin, and interpret your result. Do this by hand.

b.Use an exact method to calculate a 95% confidence interval for the true probability of heads for Kerrich’s coin (use R software), and interpret your result.

c.Compare your results from a and b. Why do the results look so similar? What would have to happen in order for these results to look substantially different?

d.Do you think the coin he used in this experiment was fair? Explain.

Answer 1

No. of of times the coin tossed =n=10000

No.of times head appears =x=5067

Probability of getting head =p x/n =5067/10000 =0.5067

Probability of not getting head = q= 1--p =1-0.5067 =0.4933

a. In case of large value of n by normal approximation probability of getting head =probability of getting trail =p=q=0.5

in such a case 95% confidence limits for true probability of getting head P given as

The confidence interval for P is ( 0.4902, 0.5098)

b, Exact method to calculate 95% confidence interval for true probability of getting head is given as

where P is the true probability of getting head

95% confidence intrval for the true probability of getting head P is (0.4969 ,0.5165)

c. When we compare the results from a and b we observe that both the intervals are more are less same. the reason behind this is the coin tossed in this experiment is almost fair

If the coin is not fair the results obtained in the above methods are quite different

d. The coin used in this experiment is fair since the probability of getting head is approximately equal to 0.5