Divide the class into two groups of approximately equal size. The students in group 1 will each toss a coin 10 times (n₁) and count the number of heads obtained. The students in group 2 will each toss a coin 40 times (n₂) and again count the number of heads. The students in each group should individually compute the proportion of heads observed, which is an estimate of p, the probability of observing a head. Thus, there will be a set of values of p1 (from group 1) and a set of values p₂ (from group 2). All of the values of p₁ and p₂ are estimates of 0.5, which is the true value of the probability of observing a head for a fair coin.(a) Which set of values is consistently closer to 0.5, the values of p₁ or p₂? Consider the proof of Theorem 3.1 on page 105 with regard to the estimates of the parameter p = 0.5. The values of p₁ were obtained with n = n₁ = 10, and the values of p₂ were obtained with n = n₂ = 40. Using the notation of the proof, the estimates are given by

where I₁, . . . , I_n1 are 0s and 1s and n₁ = 10, and

where I₁, . . . , I_n2 , again, are 0s and 1s and n₂ = 40.

(b) Referring again to Theorem 3.1, show that E(p₁) = E(p₂) = p = 0.5.

(c) Show that?

is 4 times the value of?

Then explain further why the values of p₂ from group 2 are more consistently closer to the true value, p = 0.5, than the values of p₁ from group 1.

Transcribed Image Text:

·+ In. xi_ I1+ ·…+ In, Pi = Pi п1 п1 I +.+ In2 + In2 P2 п2 n2