Transcript Estimation

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%