Question

(a) Explain what "95% confidence" means in a 95% confidence interval?

(b) What type of variable is required to construct a confidence interval for a population proportion?

(c) Explain why quadrupling the sample size causes the margin of error to be cut in half?

Answer 1

Solutions:

(a) "95% confidence" in a 95% confidence interval means that there is a 95% probability that the interval contain the true population parameter. In other words, it means that if we construct 100 different confidence interval based on 100 different samples of the same size from the same population, then we can expect that 95 of those intervals will contain the true population parameter and 5 of those will not contain the population parameter.

(b) The confidence interval for a population proportion will require a qualitative variable with two possible outcomes. For example, if we want to construct the confidence interval for a proportion of people who smoke a cigarette. Then our variable will be 'Smoking' and the two possible outcomes will be 'Yes' and 'No'. The quantity p= x/n will then give the sample proportion which is used to construct the confidence interval for the population proportion. x denotes the no. of people having a characteristic and n is the total no. of people. In this case, x denotes the no. of people who smoke, and n the total target population.

(c) The margin of error is inversely proportional to the square root of the sample size(n). Thus if we quadruple the sample size, then n will become 4n and after taking the square root this will become 2*sqrt(n). Thus, now the margin of error will be inversely proportional to 2*sqrt(n) while previously it was inversely proportional to sqrt(n), hence the margin of error will reduce to half.