Final answer:
To determine the probability or percentage of samples within a specific interval, one uses the normal distribution, calculates areas under the curve, constructs confidence intervals, and understands percentiles. The confidence interval is a range based on the sample data where the true population parameter likely lies, and it is not a direct indication of the amount of data included within that range. Confidence levels affect the width of the confidence interval, with higher confidence levels producing broader intervals.
Step-by-step explanation:
Finding the probability or percentage of samples within a specific interval involves understanding the concepts of probability and statistics, as well as utilizing tools like the normal distribution, confidence intervals, and z-scores or t-scores. For example, to find the probability that the sum of a sample is between two values, we use the normal distribution and calculate the area under the curve between these values. The confidence interval gives us a range where we believe the true population parameter lies, based on our sample data. In the context of a 90 percent confidence interval, we include the central 90 percent of the normal distribution's probability, leaving 5 percent in each tail.
To calculate specific percentiles, like the 84th and 16th, we use inverse probability techniques to find the z-scores or t-scores that correspond to these percentiles in the distribution. This will then be used to determine the sample values that fall within these percentiles. For instance, using a probability distribution, we can calculate the confidence intervals for the proportion of statistics students who use a certain product weekly, adjusting for factors like sample size and variability.
It is a common misconception that a 90 percent confidence interval contains 90 percent of the data; this is not necessarily true. The percentage of data that lies within the confidence interval does not have to match the confidence level. We can also construct confidence intervals for true proportions, such as the proportion of American adults who download music weekly. The width of these intervals can be affected by the confidence level; as it decreases, the interval generally becomes narrower.