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Estimation CJ 526 Statistical Analysis in Criminal Justice Point Estimation 1. Using a sample statistic to estimate a population parameter Example of Point Estimation 1. 2. 3. Dr. Tulip wants to know what the average age of convicted robbers in ShowMeLand is She selects a sample of 125 convicted robbers, and determines that M = 26.3 She estimates µ to be 26.3 as well Interval Estimation Constructing an interval about a sample statistic Mean plus or minus Z (alpha) * (SD/square root of N) This line above is the standard error of measurement (SEM) 1. Example of Interval Estimation 1. 2. 3. Dr. Violet wants to determine the average number of arrests that police officers make She selects a sample of 58 police officers, and calculates M = 2.3 and SEM = 1.1 She can be 68% confident that µ lies somewhere in the interval of 1.2 to 3.4 Properties of Good Estimators 1. Unbiased 1. Mean of sampling distribution is equal to the parameter being estimated Confidence Intervals and the Normal Distribution 1. 2. 95%: 1.96 99%: 2.58 Confidence Intervals for Means From Large Samples 1. 2. N > 30 M z * SEM Example of a Large Sample Confidence Interval 1. 2. 3. 4. 5. Dr. Topaz wants to know the average number of siblings that juvenile delinquents have She selects a sample of 440 juvenile delinquents, and finds M = 3.5 and SEM = 1.2 She wants to be 95% confident 3.5 ± 1.96(1.2) She can be 95% confident that µ lies somewhere in the interval of 1.148 to 5.852 Confidence Intervals and the Mean for Small Samples 1. 2. N 30 M t * SEM Example of Small Sample Confidence Interval 1. 2. 3. 4. 5. Dr. Daisy wants to know the average number of grades that juvenile delinquents fail She selects a sample of 28, and finds M = 1.4 and SEM = 0.3 She wants to be 99% confident 1.4 ± 2.771(0.3) She can be 99% confident that µ lies somewhere in the interval of 0.5687 to 2.2313 t-Distribution 1. Infinite number of curves, based on number of degrees of freedom Degrees of Freedom 1. Assume that the sum of three numbers is 10 Two of the number are 5 and 3 Degrees of Freedom -- continued 4. What can the value of X3 be? 1. It must be 2 2. It is not free to vary Confidence Intervals and Proportions for Large Samples 1. 2. N > 30 p z * SEP Standard Error of the Proportion P p(1 p) n Estimating the Standard Error of the Proportion 1. Conservative approach 1. Set p = .5 Example of Large Sample Confidence Interval for a Proportion Dr. Edna wants to determine what proportion of the general population supports the death penalty She selects a sample of 1,200, and finds p = .78 and SEP = 0.4 She wants to be 99% confident .78 ± 2.58(0.4) She can be 99% confident that P lies somewhere in the interval of .76968 to .79032 Confidence Intervals and Proportions for Small Samples 1. 2. N 30 p t * SEP Example of Small Sample Confidence Interval for a Proportion Dr. Felicia wants to determine what proportion of the general population is in favor of decriminalizing marijuana She selects a sample 26, and finds p = .34 and SEP = 0.5 She wants to be 95% confident .34 ± 2.06(0.5) She can be 95% confident that P lies somewhere in the interval of 0.327 to 0.353 Using the SPSS Explore Procedure to Generate Confidence Intervals Analyze, Descriptive Statistics, Explore Move Dependent Variable over to Dependent List Move Independent Variable over to Factor List Statistics button Set confidence level under Confidence Interval for Mean Default value is 95%