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

2. Parameter Estimation • Know X ---> what is m ? • Point estimate • single value: X and s • compute from sample • Confidence interval • range of values • probably contains m ~

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

4. Critical Value of a Statistic • Value of statistic • that marks boundary of specified area • in tail of distribution • zCV.05= ±1.96 • area = .025 in each tail ~

5. .025 .025 -2 -1 0 1 2 +1.96 -1.96 Critical Value of a Statistic f .95 z

6. Confidence Intervals • Range of values that m is expected to lie within • 95% confidence interval • P=.95 mwill fall within range • level of confidence • Which level of confidence to use? • Cost vs. benefits judgement ~

7. < m< (s X) (s X) (s X) X - zCV X + zCV X ±zCV Lower limit Upper limit or Finding Confidence Intervals • Method depends on whether s is known • If s known

8. When s Is Unknown • Usually do not know s • s is “best”point-estimator • standard error of mean for sample

9. When s Is Unknown • Cannot use z distribution • 2 uncertain values: mands • need wider interval to be confident • Student’s t distribution • normal distribution • width depends on how well s approximates s ~

10. Student’s t Distribution • if s = s, then t and z identical • if s¹ s, then t wider • Accuracy of s as point-estimate • larger n ---> more accurate • n > 120 • s»s • t and z distributions almost identical ~

11. 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 ~

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

13. 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 30 2.042 2.750 60 2.000 2.660 120 1.980 2.617 ¥ 1.96 2.576

14. (sX) X ±tCV < m < (sX) (sX) X - tCV X + tCV Lower limit Upper limit or [df = n -1] Confidence Intervals: s unknown • Same as known but use t • Use sample standard error of mean • df = n-1 [df = n -1]

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

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