Transcript Parameter Estimation

Parameter Estimation
Chapter 8
Homework: 1-7, 9, 10
Focus: when s is known (use z table)
Describing Populations
Chap 7
 Knew population ---> describe samples
 Sampling distribution of means,
standard error of the means
 Reality: usually do not know m, s

impractical

Select representative sample
 find statistics: X, s ~
Parameter Estimation
Know X ---> what is m ?
 Estimation techniques
 Point estimate
single value: X and s
 Confidence interval
range of values
probably contains m ~

Point-estimation
X is an unbiased estimator
 if repeated point-estimating infinitely...
 as many X less than m as greater than
 mode & median also unbiased
estimator of m
but neither is best estimator of m
 X is best unbiased estimator of m ~

Distribution Of Sample Means
How close is X to m?
 look at sampling distribution of means
 Probably within 2 standard errors of
mean
 about 96% of sample means
 2 standard errors above or below m
 Probably: P=.95 (or .99, or .999, etc.) ~

How close is X to m?
P(X = m + 2s)
 96%
f
-2
-1
0
1
2
Distribution Of Sample Means

If area = .95
 exactly how many standard errors
above/below m ?
 Table A.1: proportions of area under
normal curve
 look up
.475: z = 1.96
~
Critical Value of a Statistic
Value of statistic
 that marks boundary of specified area
 in tail of distribution
 zCV.05 =  1.96
 area = .025 in each tail
 5% of X are beyond 1.96
 or 95% of X fall within 1.96 standard
errors of mean ~

Critical Value of a Statistic
f
.95
.025
-2
-1.96
-1
0
.025
1
2
+1.96
Confidence Intervals
Range of values that m is expected to
lie within
 95% confidence interval
 .95 probability that m will fall within
range
 probability is the level of confidence

e.g., .75 (uncommon), or .99 or .999

Which level of confidence to use?
 Cost vs. benefits judgement ~
Finding Confidence Intervals
Method depends on whether s is
known
 If s known

X - zCV (s X) < m < X + zCV(s X)
Lower limit
or
X  zCV (s X)
Upper limit
Meaning of Confidence Interval
95% confident that m lies between
lower & upper limit
 NOT absolutely certain
 .95 probability
 If computed C.I. 100 times
 using same methods
 m within range about 95 times
 Never know m for certain
 95% confident within interval ~

Example
Compute 95% C.I.
 IQ scores
 s = 15
 Sample: 114, 118, 122, 126
 SXi = 480, X = 120, sX = 7.5
 120  1.96(7.5)
 120 + 14.7
 105.3 < m < 134.7
 We are 95% confident that population
means lies between 105.3 and 134.7 ~

Changing the Level of Confidence
We want to be 99% confident
 using same data
 z for area = .005
 zCV .01 = 2.57
 120  2.57(7.5)
 100.7 < m < 139.3
 Wider than 95% confidence interval
 wider interval ---> more confident ~

.
When s Is Unknown
Usually do not know s
 Use different formula
 “Best”(unbiased) point-estimator of
s =s
 standard error of mean for sample

s
sX 
n
When s Is Unknown
Cannot use z distribution
 2 uncertain values: m and s
 need wider interval to be confident
 Student’s t distribution
 also normal distribution
 width depends on how well s
approximates s ~

Student’s t Distribution
if s = s, then t and z identical
 if s  s, then t wider
 Accuracy of s as point-estimate
 depends on sample size
 larger n ---> more accurate
 n > 120
 s  s
 t and z distributions almost identical ~

Degrees of Freedom
Width of t depends on n
 Degrees of Freedom
 related to sample size
 larger sample ---> better estimate
 n - 1 to compute s ~

Critical Values of t
Table A.2: “Critical Values of t”
 df = n - 1
 level of significance for two-tailed test


a
area in both tails for critical value
 level of confidence for CI ~
 1 - a
~

Critical Values of t

Critical value depends on degrees of
freedom & level of significance
df
1
2
5
10
60
120
infinity
.05
12.706
4.303
2.571
2.228
2.000
1.980
1.96
.01
63.657
9.925
4.032
3.169
2.660
2.617
2.576
Critical Values of t
df = 1 means sample size is n = 2
 s probably not good estimator of s
 need wider confidence intervals
 df > 120; s  s
 t distribution  z distribution
 df > 5, moderately-good estimator
 df > 30, excellent estimator ~

Confidence Intervals: s unknown

Same as known but use t
 Use sample standard error of mean
 df = n-1
X - tCV (s X) < m < X + tCV(s X)
Lower limit
or
[df = n -1]
Upper limit
X  tCV (s X)
[df = n -1]
4 factors that affect CI width
Would like to be narrow as possible
 usually reflects less uncertainty
 Narrower CI by...
1. Increasing n
 decreases standard error
2. Decreasing s or s
 little control over this ~

4 factors that affect CI width
3. s known
 use z distribution, critical values
4. Decreasing level of confidence
 increases uncertainty that m lies
within interval
 costs / benefits ~