A confidence interval for a population proportion is a reasonable range of values where we expect the population proportion to fall within, with a chosen degree of confidence. We create confidence intervals because, as seen in previous in-class activities, sample proportions vary from sample to sample. A sample proportion will most likely not be equal to the true population parameter. Thus, instead of simply using a single value (a point estimate) to estimate our parameter, we will use a range of values. Confidence intervals take sampling variability into account and convey information on estimate accuracy. A confidence interval is calculated using the point estimate and the margin of error. The margin of error (E) is what determines the width of the interval. A confidence interval will have a width of twice the margin of error. The following is a visualization of a confidence interval. margin of error ( E ) margin of error ( E ) point estimate Confidence interval: has width 2E The margin of error is calculated using the standard error and the z critical value for the confidence level. This means that the width of our interval is determined by how much our sample data varies and how confident we want to be in our method. The confidence level, C, tells us how much confidence we have in the method used to construct the interval. It corresponds to the percentage of all intervals we would expect to contain the true population parameter. For example if we have a 95% confidence level and we took a very large number of different samples and created confidence intervals for each one, we would expect about 95% of them to contain the population parameter. For proportions, each confidence level has a corresponding z critical value (2 *). In Question 5, we applied the Empirical Rule, which indicates z* = 2 and its negative counterpart, - 2. As you can see below, the values of +2 separate the middle 95.45% from the most extreme 4.55% areas. Note that the most extreme area is split evenly in each tail.2.28% 95.45% 2.28% - -3 3 In practice, we are usually interested in 90%, 95%, and 99% confidence intervals. Below is the standard normal distribution displaying the z critical value, >*, for a 95% confidence level. As you can see, 95% of the curve is shaded in and the remaining 5% of the curve is unshaded in the tails. The values that separate the middle 95% from the most extreme 5% are what we will use for our confidence interval. 2*: The z critical value; this is the point on the standard normal distribution such that the proportion of area under the curve between - > * and + 2* is c, the confidence level. The z* value is found using the DCMP Normal Distribution tool at https://dcmathpathways.shinyapps.io/NormalDist/ 5. Select the Find Percentile/Quantile tab. . The tool defaults to a standard normal distribution with mean / = 0 and standard deviation o = 1. . Select "Two-tailed" and enter 95 for "Central Probability," since we want the middle 95 %. . The z critical value, >*, is 1.96. Note the tool presents both the z critical value and its negative counterpart, - 1.96. The Normal Distribution Explony Find Probability Find Percentile Quartile Normal Distribution with u = 0 and q = 1 Mean p: P(-1.96