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

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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

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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

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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

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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

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How close is X to ? f 120-2 96% P(X = + 2 )

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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

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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

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f 120-2 +1.96-1.96.95.025

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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 ~

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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

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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 ~

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n Compute 95% C.I. n IQ scores = 15 n Sample: 114, 118, 122, 126 X i = 480, X = 120, X = 7.5 120 1.96(7.5) l 120 + 14.7 105.3 < < 134.7 n We are 95% confident that population means lies between 105.3 and 134.7 ~ Example

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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 = 2.57 120 2.57(7.5) 100.7 < < 139.3 n Wider than 95% confidence interval l wider interval ---> more confident ~

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

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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 ~

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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 ~

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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 ~

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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 - ~

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Critical Values of t n Critical value depends on degrees of freedom & level of significance df.05.01 112.70663.657 24.3039.925 52.5714.032 102.2283.169 602.0002.660 1201.9802.617 infinity1.962.576

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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 ~

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

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4 factors that affect CI width n Would like to be narrow as possible l usually reflects less uncertainty n Narrower CI by... 1. Increasing n l decreases standard error 2. Decreasing s or l little control over this ~

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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 ~

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H1H1 H1H1 HoHo Z = 0 Two Tailed test. Z score where 2.5% of the distribution lies in the tail: Z = + 1.96 Critical value for a two tailed test.

H1H1 H1H1 HoHo Z = 0 Two Tailed test. Z score where 2.5% of the distribution lies in the tail: Z = + 1.96 Critical value for a two tailed test.

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