Parameter Estimation Chapter 8 Homework: 1-7, 9, 10 Focus: when  is known (use z table)

n Chap 7 l Knew population ---> describe samples l Sampling distribution of means, standard error of the means Reality: usually do not know  impractical n Select representative sample l find statistics: X, s ~ Describing Populations

Know X ---> what is  n Estimation techniques l Point estimate single value: X and s l Confidence interval range of values probably contains  ~ Parameter Estimation

n X is an unbiased estimator l 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  ? l look at sampling distribution of means n Probably within 2 standard errors of mean l about 96% of sample means 2 standard errors above or below  l Probably: P=.95 (or.99, or.999, etc.) ~ Distribution Of Sample Means

How close is X to  ? f   96% P(X =  + 2  )

Distribution Of Sample Means n If area =.95 exactly how many standard errors above/below  ? l Table A.1: proportions of area under normal curve l look up.475: z = ~ 1.96

n Value of statistic l that marks boundary of specified area l in tail of distribution z CV.05 =  1.96 l area =.025 in each tail l 5% of X are beyond 1.96 l or 95% of X fall within 1.96 standard errors of mean ~ Critical Value of a Statistic

f

Confidence Intervals Range of values that  is expected to lie within n 95% confidence interval.95 probability that  will fall within range l probability is the level of confidence e.g.,.75 (uncommon), or.99 or.999 n Which level of confidence to use? l Cost vs. benefits judgement ~

Finding Confidence Intervals Method depends on whether  is known If  known X - z CV  X ) X + z CV  X ) <  < X   z CV  X ) or Lower limitUpper limit

Meaning of Confidence Interval 95% confident that  lies between lower & upper limit l NOT absolutely certain l.95 probability n If computed C.I. 100 times l using same methods  within range about 95 times Never know  for certain l 95% confident within interval ~

n Compute 95% C.I. n IQ scores  = 15 n Sample: 114, 118, 122, 126  X i = 480, X = 120,  X =  1.96(7.5) l <  < n We are 95% confident that population means lies between and ~ Example

Changing the Level of Confidence n We want to be 99% confident l using same data l z for area =.005 l z CV..01 =  2.57(7.5) <  < n Wider than 95% confidence interval l wider interval ---> more confident ~

When  Is Unknown Usually do not know  n Use different formula “Best”(unbiased) point-estimator of  = s l standard error of mean for sample

When  Is Unknown n Cannot use z distribution 2 uncertain values:  and  l need wider interval to be confident n Student’s t distribution l 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 n Accuracy of s as point-estimate l depends on sample size l larger n ---> more accurate n n > 120 s   l t and z distributions almost identical ~

Degrees of Freedom n Width of t depends on n n Degrees of Freedom l related to sample size l larger sample ---> better estimate l n - 1 to compute s ~

Critical Values of t n Table A.2: “Critical Values of t” n df = n - 1 n level of significance for two-tailed test  l area in both tails for critical value n level of confidence for CI ~ 1 -  ~

Critical Values of t n Critical value depends on degrees of freedom & level of significance df infinity

Critical Values of t n df = 1 means sample size is n = 2 s probably not good estimator of  l need wider confidence intervals df > 120; s   t distribution  z distribution l df > 5, moderately-good estimator l df > 30, excellent estimator ~

Confidence Intervals:  unknown n Same as known but use t l Use sample standard error of mean l df = n-1 X - t CV sX)sX) X + t CV sX)sX) <  < Lower limitUpper limit [df = n -1] X   t CV sX)sX) or [df = n -1]

4 factors that affect CI width n Would like to be narrow as possible l usually reflects less uncertainty n Narrower CI by Increasing n l decreases standard error 2. Decreasing s or  l little control over this ~

4 factors that affect CI width 3.  known l use z distribution, critical values 4. Decreasing level of confidence increases uncertainty that  lies within interval l costs / benefits ~