Question

What percent of sample proportions results in a 75% confidence interval that includes the population proportion?

a) 75%
b) 25%
c) 50%
d) 100%

Answer 1

Seventy-five percent of sample proportions result in a 75% confidence interval that includes the population proportion. Option A is answer.

Step-by-step explanation:

When constructing a confidence interval for a population proportion, we are trying to estimate the true proportion within a certain margin of error. The level of confidence indicates the probability that the interval will actually contain the true proportion. A 75% confidence interval means that there is a 75% chance that the interval will include the true proportion.

In other words, if we were to construct many 75% confidence intervals for the same population proportion, we would expect that 75% of those intervals would contain the true proportion. This is because the 75% confidence level is based on the normal distribution, which tells us that 75% of the data falls within 1.96 standard deviations of the mean.

Therefore, the answer is 75%. Option A is answer.

Answer 2

When interpreting a 75% confidence interval, 75% of these intervals from repeated samples will include the true population proportion.

Step-by-step explanation:

The student is asking about the confidence interval and what percentage of sample proportions would result in a confidence interval that includes the actual population proportion. When constructing a 75% confidence interval, it means that if we were to take many samples and build a confidence interval from each of them, 75% of those intervals would contain the true population proportion. Therefore, the correct answer is a) 75%. This is a fundamental concept in statistics which asserts that a confidence interval includes the population parameter a certain percentage of times in many repeated samples. The common misunderstanding is thinking the confidence level represents the probability that a single confidence interval contains the population parameter, but it actually refers to the long run proportion of such intervals that will include the parameter if the process is repeated indefinitely.