Mass Incarceration In Depth
Delve into the complex issues surrounding mass incarceration, explore international comparison data, study the impact of gender on incarceration rates, and test your knowledge on key statistics in the criminal justice system.
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Unit 3: Foundations forinference 2. Confidenceintervals Sta 101 Fall 2018 Duke University, Department of Statistical Science Dr. Ellison Slides posted at https://www2.stat.duke.edu/courses/Fall18/sta101.001/
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
Announcements Peer evalfeedback due Tuesday 9/25 11:59pm: Part of your participation grade. Lateness / absence, mental presence, communication, commitment Tips: Move seats around in class and lab, give your best, and don t assume the worst Instructions: Give scores that reflect your verbal comments Midterm 1 review Wednesday 9/26 Lab Assignment 3 due Thursday 9/27 before your section time. Midterm 1 Monday October 1 After Midterm Project Explanation Problem Set 3 due Friday October 5 11:55pm 1
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
1. Statistical inference methods based on the CLT depend on thesame conditions as the CLT Always check these in context of the data and the research question! 1. Independence: Sampled observations must be independent. * This is difficult to verify, but is more likely if random sampling/assignment is used, and, if sampling without replacement, n < 10% of thepopulation. 2. Sample size/skew: 1. Either the population distribution is normal OR 2. n > 30 and the population distribution is not extremely skewed (the more skewed the distribution, the higher n necessary for the CLT toapply). 2
1. Statistical inference methods based on the CLT depend on thesame conditions as the CLT Always check these in context of the data and the research question! 1. Independence: Sampled observations must be independent. * This is difficult to verify, but is more likely if random sampling/assignment is used, and, if sampling without replacement, n < 10% of thepopulation. 2. Sample size/skew: 1. Either the population distribution is normal OR 2. n > 30 and the population distribution is not extremely skewed (the more skewed the distribution, the higher n necessary for the CLT toapply). we know? How do 2
1. Statistical inference methods based on the CLT depend on thesame conditions as the CLT Always check these in context of the data and the research question! 1. Independence: Sampled observations must be independent. * This is difficult to verify, but is more likely if random sampling/assignment is used, and, if sampling without replacement, n < 10% of thepopulation. 2. Sample size/skew: 1. Either the population distribution is normal OR 2. n > 30 and the population distribution is not extremely skewed (the more skewed the distribution, the higher n necessary for the CLT toapply). *This is also difficult to verify for the population, but we can check it using the sample data, and assume that the sample mirrors the population. 2
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
2. Use confidence intervals to estimate population parameters CI : point estimate margin of error 3
2. Use confidence intervals to estimate population parameters CI : point estimate margin of error CI : point estimate (critical value)(standard error) 3
2. Use confidence intervals to estimate population parameters Case 1: You know population standard deviation . ?? ??? ? = ? ? ? 1. Independence: Sampled observations must be independent. * This is difficult to verify, but is more likely if random sampling/assignment is used, and, if sampling without replacement, n < 10% of the population. 2. Sample size/skew: 1. Either the population distribution is normal OR 2. n > 30 and the population distribution is not extremely skewed (the more skewed the distribution, the higher n necessary for the CLT to apply). ?
2. Use confidence intervals to estimate population parameters Case 2: You don t know population standard deviation . ?? ??? ? = ? ? ? 1. Independence: Sampled observations must be independent. * This is difficult to verify, but is more likely if random sampling/assignment is used, and, if sampling without replacement, n < 10% of the population. 2. Sample size/skew: 1. Either the population distribution is normal OR 2. n > 30 and the population distribution is not extremely skewed (the more skewed the distribution, the higher n necessary for the CLT to apply). ?
2. Use confidence intervals to estimate population parameters What s missing? How do you construct a confidence interval when you don t know population standard deviation and n < 30 and the population is approximately normally distributed? (unit 4) How do you construct a confidence interval when n < 30 and the population and the population is NOT approximately normally distributed? ( )
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
3. Critical value depends on the confidence level Clicker question What is the critical value (Z ) for a confidence interval at the 91% confidence level? (a) Z = 1.34 (b) Z = 1.65 (c) Z = 1.70 (d) Z = 1.96 (e)Z = 2.33 4
3. Critical value depends on the confidence level Clicker question What is the critical value (Z ) for a confidence interval at the 91% confidence level? (a) Z = 1.34 (b) Z = 1.65 (c) Z = 1.70 (d) Z = 1.96 (e)Z = 2.33 0.91 0.045 0.045 4
3. Critical value depends on the confidence level Clicker question What is the critical value (Z ) for a confidence interval at the 91% confidence level? Standard Normal Distribution Z~N(0,1) (a) Z = 1.34 (b) Z = 1.65 (c) Z = 1.70 (d) Z = 1.96 (e)Z = 2.33 0.91 0.045 0.045 0 4
3. Critical value depends on the confidence level Clicker question What is the critical value (Z ) for a confidence interval at the 91% confidence level? Standard Normal Distribution Z~N(0,1) (a) Z = 1.34 (b) Z = 1.65 (c) Z = 1.70 (d) Z = 1.96 (e)Z = 2.33 0.91 0.045 0.045 1.7 0 1.7 4
1 % ?????????? ???????? = (????? ????????) ? (standard error) ?2 *if using the z- distribution Standard Normal Distribution Z~N(0,1) 1- /2 /2 ? ? ?2 ?2
Common misconceptions about confidence intervals 1. The confidence level of a confidence interval is the probability that the true population parameter is in the confidence interval you construct for a single sample. The confidence level is equal to the proportion of random samples that result in confidence intervals that contain the true pop. parameter. 5
Common misconceptions about confidence intervals 1. The confidence level of a confidence interval is the probability that the true population parameter is in the confidence interval you construct for a single sample. The confidence level is equal to the proportion of random samples that result in confidence intervals that contain the true pop. parameter. 2. A narrower confidence interval is always better. This is incorrect since the width is a function of both the confidence level and the standard error. 5
Common misconceptions about confidence intervals 1. The confidence level of a confidence interval is the probability that the true population parameter is in the confidence interval you construct for a single sample. The confidence level is equal to the proportion of random samples that result in confidence intervals that contain the true pop. parameter. 2. A narrower confidence interval is always better. This is incorrect since the width is a function of both the confidence level and the standard error. 3. A wider interval means less confidence. This is incorrect since it is possible to make very precise statements with very little confidence. 5
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
4. Calculate the sample size a priori to achieve desired margin oferror So if we know the desired ME, and confidence level (andhence z ), and the sample standard deviation, we can solve for n. 6
Application exercise: 3.1 Confidence interval for a singlemean See course website for details. 7
Outline 1. Housekeeping 2. Main ideas 1. depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error Statistical inference methods based on the CLT 3. Summary
Summary of mainideas 1. Statistical inference methods based on the CLT depend on the same conditions as the CLT 2. Use confidence intervals to estimate population parameters 3. Critical value depends on the confidence level 4. Calculate the sample size a priori to achieve desired margin of error 8
