Summary Stats 188 chapter 8 notes

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATION

> CHARACTERISTICS OF ESTIMATORS :



◦ unbiased ness :
expectation of the estimator equals the parameter value .




◦
Consistency : Unbiased estimator that approaches population values as sample size increases .




Efficiency Unbiased estimator that has minimal standard error
◦ : .




> A confidence interval estimate = a
range of numbers ,
called an interval ,
constructed around the point estimate




GENERAL FORMULA : A point estimate is the


value of a single sample


Point Estimate ± ( critical value ) ( standard Error ) statistic ,
such as the mean
,




CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN 10 Known )

◦ ASSUMPTIONS :




population standard deviation 0 IS known



population is
normally distributed


If population is not normal ,
use large sample In 30 )




CONFIDENCE INTERVAL ESTIMATE :



-




± Za, , O : point estimate
n


✗
12
: the normal distribution critical Value for a
probability Of ✗ 12 In each tail


" " " "" " ""




EXAMPLE used
:
commonly confidence
◦




1 Consider a 95.1 .
confidence interval : £42 = I 1.96 intervals are : 90 1.
-




,
95 1
'



.




I ✗ 0.95 ✗ and 99.1
'
-
= . . = 0.05 .




'0
I >




↑
a

2=0 =
0.025



.
V




ZX / 2=-1.96 O ZX / 2=1.96
2- units :




✗ Units : Lower confidence point upper confidence
Limit Estimate Limit

, >
SAMPLING ERROR :


sampling due single sample from the population
•

is the variation
error that occurs to
selecting a .




>
The size of the sampling error =
primarily based on the amount of variation in the population and on the


sample size .




>
Large samples have less large to select
sampling error than small samples , but samples cost more .




* The value of 2- ✗ 12 needed to construct a confidence interval is called the critical value for the distribution




CONFIDENCE INTERVAL ESTIMATE FOR THE MEAN 10 unknown )




o If the population standard deviation 0 Is unknown
,
we can substitute the sample standard deviation ,
S



1 This introduces an extra uncertainty ,
since S Is a variable from sample to sample .




> distribution normal distribution
'


. . Use the t instead Of the




o ASSUMPTIONS :




population standard deviation is unknown


population is normally distributed




CONFIDENCE INTERVAL ESTIMATE i




s
t ✗ 12
-




n




>
STUDENT 'S T DISTRIBUTION NOTE : t > 2- as n increases


• The tan value depends on degrees of freedom Idf )


df = n -

l standard normal




fi)
Lumber Of Observations that are free to (t with df =
X )


vary after sample mean has been calculated

no , ,},




g.) [ tldf
(
= 5)

t