Parameter Estimation

Chapter 8

Homework: 1-7, 9, 10

Focus: when is known (use z table)

Chap 7 Knew population ---> describe samples Sampling distribution of means,

standard error of the means Reality: usually do not know

impractical Select representative sample

find statistics: X, s ~

Describing Populations

Know X ---> what is Estimation techniques

Point estimate

single value: X and s Confidence interval

range of values

probably contains ~

Parameter Estimation

X is an unbiased estimator if repeated point-estimating infinitely... as many X less than as greater than mode & median also unbiased

estimator of

but neither is best estimator of X is best unbiased estimator of ~

Point-estimation

How close is X to ? look at sampling distribution of means

Probably within 2 standard errors of mean about 96% of sample means 2 standard errors above or below Probably: P=.95 (or .99, or .999, etc.) ~

Distribution Of Sample Means

How close is X to ?

f

1 20-1-2

96%

P(X = + 2)

Distribution Of Sample Means

If area = .95 exactly how many standard errors

above/below ? Table A.1: proportions of area under

normal curve look up

.475: z = ~1.96

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

Critical Value of a Statistic

f

1 20-1-2

+1.96-1.96

.95 .025.025

Confidence Intervals

Range of values that is expected to lie within

95% confidence interval .95 probability that 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 is known

If known

X - zCV X) X + zCVX)< <

X zCVX)or

Lower limit Upper limit

Meaning of Confidence Interval

95% confident that lies between lower & upper limit NOT absolutely certain .95 probability

If computed C.I. 100 times using same methods within range about 95 times

Never know for certain 95% confident within interval ~

Compute 95% C.I. IQ scores

= 15 Sample: 114, 118, 122, 126

Xi = 480, X = 120, X = 7.5 120 1.96(7.5) 120 + 14.7 105.3 < < 134.7

We are 95% confident that population means lies between 105.3 and 134.7 ~

Example

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 < < 139.3

Wider than 95% confidence interval wider interval ---> more confident ~

When Is Unknown

Usually do not know Use different formula “Best”(unbiased) point-estimator of

= s standard error of mean for sample

n

ssX

When Is Unknown

Cannot use z distribution 2 uncertain values: and need wider interval to be confident

Student’s t distribution also normal distribution width depends on how well s

approximates ~

Student’s t Distribution

if s = , then t and z identical if s , then t wider

Accuracy of s as point-estimate depends on sample size larger n ---> more accurate

n > 120 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

area in both tails for critical value

level of confidence for CI ~ 1 - ~

Critical Values of t

Critical value depends on degrees of freedom & level of significance

df .05 .01

1 12.706 63.657

2 4.303 9.925

5 2.571 4.032

10 2.228 3.169

60 2.000 2.660

120 1.980 2.617

infinity 1.96 2.576

Critical Values of t

df = 1 means sample size is n = 2 s probably not good estimator of need wider confidence intervals

df > 120; s t distribution z distribution df > 5, moderately-good estimator df > 30, excellent estimator ~

Confidence Intervals: unknown

Same as known but use t Use sample standard error of mean df = n-1

X - tCV sX) X + tCVsX)< <

Lower limit Upper limit

[df = n -1]

X tCVsX)or [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 little control over this ~

4 factors that affect CI width

3. known use z distribution, critical values

4. Decreasing level of confidence increases uncertainty that lies

within interval costs / benefits ~