Central Limit Theorem

Text of section.

Suppose a coin has probability 0.4 of coming up heads, and we flip the coin 100 times. Let S be the number of heads. Estimate the probability that 34 \leq S \leq 44.

Suppose a fair die is rolled 100 times, and let m be the average value of the rolls. Estimate the probability that 3.45 \leq m \leq 3.55.

The heights of men in the US have a mean of 69 in and a variance of about 9 in^2, and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in^2.

Suppose the heights of 30 men are sampled, and a sample mean m is taken. Find the expected value and variance of m.

Estimate the probability that m \geq 69.5.

What if the sampled group was women?