Final answer:
A precision interval for a sampling risk of 10 percent that ranges from 60 to 70 indicates the estimated range where the sample statistic likely resides, considering the margin of sampling error. A 90 percent confidence interval is narrower than a 95 percent one because it accepts a greater risk of leaving the true parameter outside the interval. As the confidence level increases, the width of the confidence interval also increases to ensure greater certainty of encompassing the true population parameter.
Step-by-step explanation:
When discussing a precision interval for a sampling risk of 10 percent that ranges from 60 to 70, we are addressing the range within which we can be confident that the sample statistic falls, given the risk of sampling error. This concept is fundamentally rooted in the realm of statistics, particularly in relation to confidence intervals. A precision interval (similar to a confidence interval) gives us an estimated range for a statistic that is likely to include the true population parameter.
For instance, the 90 percent confidence interval is typically narrower than the 95 percent confidence interval because it leaves a smaller portion (10 percent) outside the interval. This means that the 90 percent confidence interval is calculated to include the central 90 percent of the probability distribution, with 5 percent in each tail, when considering a normal distribution. To illuminate this principle, we can look at an example where the 90 percent confidence interval is (67.18, 68.82) and compare it to a 95 percent confidence interval which is wider, such as (67.02, 68.98). This is because to be 95 percent confident, the interval must be wider to account for the reduced risk of the true parameter not being within the interval.
When confidence levels increase, confidence intervals become wider. If we were to create a 99 percent confidence interval for a particular population parameter, it would be wider than both the 90 percent and 95 percent intervals, reflective of a greater certainty in containing the true population value. Factors such as sample size and variability affect the width of the confidence interval as well; larger sample sizes and lower variability generally result in narrower intervals.