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Parameter Estimation Chapter 8 Homework: 1-7, 9, 10

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Know X ---> what is n Point estimate l single value: X and s l compute from sample n Confidence interval l range of values probably contains ~ Parameter Estimation

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How close is X to ? l look at sampling distribution of means Probably within 2 X l Use: P=.95 u or.99, or.999, etc. ~ Parameter Estimation

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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 ~ Critical Value of a Statistic

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120-2 f z.95.025 +1.96-1.96

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Confidence Intervals Range of values that is expected to lie within n 95% confidence interval P=.95 will fall within range l level of confidence 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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When Is Unknown Usually do not know n s is “best”point-estimator 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 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 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 total 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 52.5714.032 102.2283.169 302.0422.750 602.0002.660 1201.9802.617 1.962.576

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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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Examples: Confidence intervals n What is population mean for high school GPA of Coe students? If unknown? l X = 3.3 s =.2n = 9 n What if n = 4? n 99% CI ?~

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Factors that affect CI width n Would like to be narrow as possible 1. Increasing n l decreases standard error l increases df 2. Decreasing s or l little control over this 3. known 4. Decreasing level of confidence l increases uncertainty ~

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