# Central Limit Theorem
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## Section 4.1 Central Limit Theorem

Text of section.
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### Exercises Exercises

#### 1.

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 \(34 \leq S \leq 44\text{.}\)
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#### 2.

Suppose a fair die is rolled 100 times, and let \(m\) be the average value of the rolls. Estimate the probability that \(3.45 \leq m \leq 3.55\text{.}\)
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#### 3.

The heights of men in the US have a mean of 69 in and a variance of about 9 in\(^2\text{,}\) and the heights of women in the US have a mean of 63.5 in with a variance of 6.25 in\(^2\text{.}\)
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##### (a)

Suppose the heights of 30 men are sampled, and a sample mean \(m\) is taken. Find the expected value and variance of \(m\text{.}\)
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##### (b)

Estimate the probability that \(m \geq 69.5\text{.}\)
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##### (c)

What if the sampled group was women?
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