Final answer:
A 95% confidence interval will indeed contain more data points than an 80% confidence interval because it is wider, reflecting a higher level of certainty that the interval captures the true population parameter. A 99% confidence interval is wider still, for the same reason. There's a trade-off between confidence level and precision of the interval estimate.
Step-by-step explanation:
The statement that a 95% confidence interval will always have more data points than an 80% confidence interval is true. This is because a confidence interval's range is related to the degree of certainty we have that the interval includes the true parameter (like the mean) of the population. A 95% confidence interval is wider than an 80% confidence interval because it is designed to include 95% of all possible outcomes, thus capturing more data points and having a wider range.
When we compare a 95% confidence interval to a 99% confidence interval, the 99% interval will be even wider because it aims to include 99% of all possible outcomes. Every increase in the confidence level will increase the width of the confidence interval, as it reflects a greater area under the normal distribution curve to capture the true population parameter.
For example, if the 90% confidence interval for a certain parameter is (67.18, 68.82), then increasing the confidence to 95% would result in a wider interval, such as (67.02, 68.98). The extra 5% confidence means that the tails of the distribution extend further, thus increasing the range of the interval to capture the population mean with greater certainty.
It is important to note that while higher confidence levels do lead to wider intervals, this also means the estimated range of the true population parameter is less precise. Therefore, there is a trade-off between the level of confidence and the precision of the interval estimate.