Download Understanding Confidence Intervals: A 95% Capture Rate and more Study Guides, Projects, Research Statistics in PDF only on Docsity! 6.2: Making sense of Confidence Intervals CP Stats 2016-2017 Some things to know about confidence intervals… A confidence level tells us that a given interval captures the parameter (for example) 95% of the time. In other words, if we were to take many, many random samples and construct a 95% confidence interval using each sample, about 95% of those would capture p. Confidence intervals are based on the model of sampling distributions that we covered last unit. Solution Example 1 If the Pew Project were to select many random samples of U.S. adults and constructed a 95% confidence interval using each sample, about 95% of all the intervals would capture the true proportion of people who use some form of social media to share updates about yourself or to see updates about others. To demonstrate the interpretation of confidence levels, let’s play around with some confidence intervals I’ve emailed you the link below. http://digitalfirst.bfwpub.com/stats_apple t/stats_applet_4_ci.html Play around with the app, and be ready to summarize: 1. Explain how changing the confidence level affects the confidence interval. 2. Explain how changing the sample size affects the length of the confidence interval. 3. Does increasing the sample size increase the capture rate (percent hit)? Example 2 Solutions Back to the Pew Internet and American Life Project reporting the 95% confidence interval for the proportion of all U.S. adults who use social media… 1. Explain what would happen to the length of the interval if the confidence level were increased to 99%. The confidence interval will be wider because increasing the confidence level increases the margin of error. 2. Explain what would happen to the length of the original interval if the sample size increased to 5000. The confidence interval will be narrower because increasing the sample size decreases the margin of error. What the margin of error does not account for We create intervals for our estimates because we anticipate the value of the point estimate to be different than the actual population mean or population proportion. The margin of error, that wiggle room, accounts for this variability we expect; it does not account f or possible bias. Example 3 As part of a project about response bias, you survey a random sample of 25 students from your school. One of the questions required students to state their GPA aloud. Based on the responses, you conclude that you are 90% confident that the interval from 3.14 to 3.52 captures the mean GPA for all students at your school. Describe one potential source of bias in your study that is not accounted for by the margin of error. Calculating a confidence interval Generally, the confidence interval for estimating a population parameter has the form Statistic ± (critical value) × (standard deviation of statistic) The critical value basically is the number of standard deviations that makes the interval wide enough to have the stated capture rate. The product of the critical value and standard deviation is the margin of error. Before calculating a confidence interval … These are the conditions you are expected to check before calculating a confidence interval for a population proportion p 1. Random: The data come from a well- designed random sample or randomized experiment. 1. Large counts: Both np-hat and n(1-phat) are at least 10.
