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1. How large should the size of a random sample be so that we can be 90% certain that the sample mean X will not deviate from the true mean by more than σ/2?

2. Let a fair coin be tossed n times and let Sn be the number of heads that turn up. Show that the fraction of heads, Sn/n, will be near to 1/2 for large n. What can we conclude if the coin is not fair?

3. Suppose that a failure of certain component follows the distribution f(x) = px(1 - p)x for x = 0, 1, and zero, elsewhere. How many components must one test in order that the sample mean X will lie within 0.4 of the true state of nature with probability at least as great as 0.95?

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