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. Dr. Tulip wants to know what the average age of convicted robbers in ShowMeLand is

2. She selects a sample of 125 convicted robbers, and determines that M = 26.3

3. She estimates µ to be 26.3 as well

Interval Estimation

1. Constructing an interval about a sample statistic

Mean plus or minusZ (alpha) * (SD/square root of N)This line above is the standard error

of measurement (SEM)

Example of Interval Estimation

1. Dr. Violet wants to determine the average number of arrests that police officers make

2. She selects a sample of 58 police officers, and calculates M = 2.3 and SEM = 1.1

3. She can be 68% confident that µ lies somewhere in the interval of 1.2 to 3.4

Properties of Good Estimators

1. Unbiased1. Mean of sampling distribution is equal

to the parameter being estimated

Confidence Intervals and the Normal Distribution

1. 95%: 1.962. 99%: 2.58

Confidence Intervals for Means From Large Samples

1. N > 302. M z * SEM

Example of a Large Sample Confidence Interval

1. Dr. Topaz wants to know the average number of siblings that juvenile delinquents have

2. She selects a sample of 440 juvenile delinquents, and finds M = 3.5 and SEM = 1.2

3. She wants to be 95% confident4. 3.5 ± 1.96(1.2)5. 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. N 302. M t * SEM

Example of Small Sample Confidence Interval

1. Dr. Daisy wants to know the average number of grades that juvenile delinquents fail

2. She selects a sample of 28, and finds M = 1.4 and SEM = 0.3

3. She wants to be 99% confident4. 1.4 ± 2.771(0.3)5. 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 22. It is not free to vary

Confidence Intervals and Proportions for Large

Samples1. N > 302. p z * SEP

Standard Error of the Proportion

n

ppP

)1(

Estimating the Standard Error of the Proportion

1. Conservative approach1. 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

Samples1. N 302. 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%