Question

3. (20 points) A box contains a large number of red and blue marbles, but the proportions are unknown; a simple random sample of 100 marbles is drawn from the box, and 48 turn out to be red. The percentage of red marbles in the box ís estimated as 48%, with an SE of 5% (you don't need to check this, assume it is accurate). Say whether each of the following is true or false, and explain briefly a. The percentage of reds in the sample will be around 48%, give or take 5% or so. b. The 48% is likely to be off the percentage of red marbles in the box, by 5% or so. c. A 95%-confidence interval for the percentage of reds in the box is 38% to 58%. d. If another sunple is taken, there is about a 95% chance that the percentage of reds in the new sample will be in the range from 38% to 58%. e. If we double the size of the sample, the corresponding confidence interval will cover a range twice as wide as the confidence interval above.

Answer 1

a) False

Because we know with certainty that the percentage of red marbles in the sample is 48%.

b) True

c) True

Since the sampling distribution of the population percentage follows the normal distribution, a 95% confidence interval for the population population parameter can be built considering about 2 (1.96) standard deviations away from the estimated parameter. In this case 2 standard deviations means about 10%.

d) True

e) False

Increasing the sample size, decreases the width of confidence intervals.