Fairness of a coin. You are given a coin and asked whether the coin is fair. That is, does the coin indicate heads 50% of time and tails the other 50%? The first 11 flips of the coin give the sequence...

Fairness of a coin. You are given a coin and asked whether the coin is fair. That is, does the coin indicate heads 50% of time and tails the other 50%? The first 11 flips of the coin give the sequence

which contains 5 heads (H) and 6 tails (T ).

(a) Can you say, with 95% confidence, that this coin is nearly fair (49% to 51% chance of heads)? If not, is there any property of the coin you can say with the same confidence? Since the number of degrees of freedom in this example is 11 − 1 = 10, the Student’s t-distribution in Table 24.1 will be helpful. It may also be useful to assign a value of 1 to a flip of heads and 0 to tails.

(b) If you flip the coin 90 more times (101 total samples) and see 36 more heads (41 total) can you make a stronger statement about the fairness of this coin?