Download Interval Estimation

CHAPTER 7 SUMMARY
CENTRAL LIMIT THEOREMS FOR x and p s :
Main point of Central Limit Theorems: Any statistic that comes from a random sample
that is large enough so that the “law of large numbers” applies will possess three characteristics:
1. Consistency: The statistic will have equal probability of understating or overstating the
actual value of the population parameter it is estimating.
2. Unbiasedness: On average, and with highest probability, the statistic will assume the
correct value of the population parameter it is estimating.
3. Efficiency: The larger the sample used to compute the statistic estimating the relevant
population parameter, the smaller the error dispersion (standard error) of the statistic.
Thus we can state three Central Limit Theorem versions (A, B, and C)for the statistics x
and
ps :
Statistic
x
ps
Consistency
Case A: Normal
distribution, when  is
known
Case B: t distribution
with (n-1) degrees of
freedom, when  is
unknown and we must
use s as an
estimate/proxy.
Case C:
Normal Distribution
Unbiasedness
x    x
x    x
 p  p  ps
s
Efficiency

x 
n
x 
p 
s
s
n
p(1  p)

n
p s (1  p s )
n
CHAPTER 8: Interval Estimation SUMMARY
OBJECTIVES:
1. To construct interval estimates for population means, , and population proportions, p, using
randomly selected sample information, along with sampling distributions.
2. To determine proper sample sizes, n, for statistical inference about populations from
samples.
Parameter
Best
Point
Estimate


p
x
ps
Interval estimate
x –E<< x +E
ps – E < p < ps + E
Margin of
Error for n
large enough
E = z· x ;
( known)
z from table
E.2 Case A
E = z· p s
(n p s >5 and
n 1  p s >5)
z from table
E.2 Case C
Margin of
Error for n
not large
enough
E = t· x
( unkown)
t from table
E.3 Case B
Procedure
cannot be
applied to
p s when
sample is not
large enough
Proper
sample
size for
inference
n=(z/E)2
n=pq(z/E)2
Four (4) steps to calculate Confidence Intervals:
1. Estimate sample statistics from sample and sampling distributions (n, x or p s , s, and  x
or  p s ).
2. Use best point estimate, x or p s , as interval midpoint.
3. Calculate the number of standard deviations you will deviate from midpoint, using
confidence level (z) and sampling distribution information ( x or  p s ); i.e., calculate
the "maximum error", E = z· x or t· x , in the case of means. Or E = z· p s , in the
case of proportions.
TO CALCULATE z: when sample size is "large enough" (table E.2)
 Divide the complement of the confidence level by 2.
 Find the above area in table E.2 and read it “inside-out”.
 This identifies the (negative) number of standard deviations, z-score, associated with
the desired confidence level.
TO CALCULATE t: when sample size is not "large enough" (table E.3)
 Divide the complement of the confidence level by 2. This determines the table E.3
column to use.
 Count degrees of freedom (df = n - 1). This determines the table E.3 row to use.
 Read the t score "outside-in".
4. To create interval estimates we add and subtract the error allowance, E, from the best
point estimate, x or p s .