Final answer:
Confidence intervals represent a range of values estimated to contain the true population parameter, and they require conditions such as plausible independence, randomization, the 10% condition, and the success/failure condition. Each percentage (90%, 95%, or 99%) indicates the amount of probability included. Misunderstandings can lead to incorrectly interpreting the meaning of a confidence interval.
Step-by-step explanation:
Understanding Confidence Intervals in Statistics, Confidence intervals are a range of values used to estimate the true value of a population parameter. This range is calculated from sample data and a chosen level of confidence. The question refers to the conditions required for constructing confidence intervals correctly:
Plausible Independence Condition - assumes that sampled observations must be independent of one another.
Randomization Condition - ensures that the data is collected in a way that is randomly selected from the population.
10% Condition - considers that the sample size should be no more than 10% of the population to avoid over-sampling.
Success/Failure Condition - requires that the sample has at least 5 successes and 5 failures for binomial distributions.
These conditions support the validity of the confidence interval, whether it's a 90%, 95%, or 99% confidence interval. Each percentage signifies how much of the probability the interval contains, excluding the remainder evenly between the tails of the distribution.
For instance, a 99 percent confidence interval contains 99 percent of the probability, meaning only 1 percent is left out, split between the upper and lower tails. A 95 percent confidence interval contains 95 percent of the probability, with 2.5 percent excluded from each tail.
Similarly, a 90 percent confidence interval includes the central 90 percent, excluding 5 percent (or 2.5 percent in each tail). If the confidence level is lowered, the error bound or margin of error decreases. This is visible when comparing a 99 percent confidence interval, with a wider range, to a 90 percent confidence interval, which will be narrower.
It's a common misconception that a 90% confidence interval includes 90% of the data; it actually means that we expect 90% of such constructed intervals to contain the true population parameter, like the mean.