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

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

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

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

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

120-2 f z.95.025 +1.96-1.96

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 ~

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

When  Is Unknown Usually do not know  n s is “best”point-estimator 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 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 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 total 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.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

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]

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 ?~

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 ~