Confidence Intervals

Text before the first section.

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?

Confidence Intervals

Text of section.

Suppose we flip a coin 100 times and count 60 heads. Let p be the (unknown) probability that the coin comes up heads on a flip. Give an approximate 95% confidence interval for the value of p.

Suppose in a sample of 100 people, 12 are left-handed.

Give a 95% confidence interval for the proportion p of left-handed people.

Give a 90% confidence interval.

The weights of five mice are measured and recorded below. Give a 95% confidence interval for the sample mean weight of mice. (Pretend 5 measurements is large enough for the CLT to apply.)

mouse i 1 2 3 4 5 weight \widetilde{w}_i (g) 26 32 33 20 29

The heights of five plants are measured and recorded below. Give a 95% confidence interval around the sample mean for the heights of the plants. (Pretend 5 measurements is large enough for the CLT to apply.)

plant i 1 2 3 4 5 height \widetilde{h}_i (in) 15 14 18 21 17