Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Probability and Statistics LECTURE 7 INTERVAL ESTIMATION Outline 1. State What Is Estimated 2. Distinguish Point & Interval Estimates 3. Explain Interval Estimates 4. Compute Confidence Interval Estimates for Population Mean & Proportion Adapted from http://www.prenhall.com/mcclave 7-1 7-2 Statistical Methods Estimation Process Population Mean, , is unknown Random Sample I am 95% confident that is between 40 & 60. Mean X= 50 Sample 7-3 Parameter vs sample statistic • Parameter is numerical descriptive measure of population • Sample statistic is a numerical descriptive measure of sample • Example 7-4 Unknown Population Parameters Are Estimated Estimate Population Parameter... Mean Proportion p Variance Differences 7-5 7-6 2 1 - 2 with Sample Statistic x p^ s 2 x1 -x2 Estimation Methods Estimation Point Estimation Interval Estimation 7-7 Point Estimation 1. Provides Single Value 2. Example: 3. Disadvantage: Gives No Information about How Close Value Is to the Unknown Population Parameter 7-8 Key Elements of Interval Estimation Interval Estimation 1. Provides Range of Values 2. Gives Information about Closeness to Unknown Population Parameter A probability that the interval contains the population parameter. Sample statistic Confidence interval (point estimate) Stated in terms of Probability Knowing Exact Closeness Requires Knowing Unknown Population Parameter 3. Example: Unknown Population Mean Lies Between 50 & 70 with 95% Confidence 7-9 Confidence limit (lower) Confidence limit (upper) 7 - 10 Confidence Interval (CI) Estimates Confidence Interval for Population Mean ( Known) 1. Assumptions Confidence Intervals Mean Known Proportion Unknown Population Standard Deviation Is Known Population Is Normally Distributed If Population Is Not Normal, Large Sample Size (so that the CLT holds) Is Required (In This Case The Sampling Distribution Of Is Approximately Normal) 2. Confidence Interval Estimate 7 - 11 7 - 12 Proof of the Confidence Interval (CI) formula Proof of the Confidence Interval (CI) formula This proof is optional Start with sampling distribution of sample mean. Recall that under some conditions, this distribution is normal or approx. normal Convert to Z distribution 1-α α/2 x 1-α α/2 Z 7 - 13 7 - 14 Proof of the Confidence Interval (CI) formula Let’s define a new notation: • Zα/2 is the Z value such that the area to its right is equal to α/2 • So we have figure: Proof of the Confidence Interval (CI) formula P(-Zα/2 < Z < Zα/2) = ? Replace Rearranging the expression, we should obtain an probability equation for the CI Z 7 - 15 -Zα/2 0 Zα/2 7 - 16 Confidence Level 1. Probability that the Interval contains Unknown Population Parameter Z and /2 /2 1- 2. Denoted 1 - Is Probability That Interval does Not contain Parameter 3. Typical Values Are 99%, 95%, 90% 7 - 17 0 As is the probability that the interval does not contain , we construct an interval that places area /2 in each tail. is the z value such that the area /2 lies to its right. 7 - 18 Z and How to look up Table 1 /2 /2 1- 0 If we know , we can find Appendix of your textbook. using Table 1 in Example: = 0.1 → /2 = 0.05 → P(Z < Look up table 1, and we find = 1.645 ) = 0.95 7 - 19 7 - 20 Factors Affecting Interval Width Z and Note that we actually find 2 values for Z by looking up table 1: With P = 0.9495, we have Z = 1.64 With P = 0.9505, we have Z = 1.65. Taking the average of the two values, we have = 1.645. 1. Data variability measured by 2. Sample Size Intervals Extend from X - ZX toX + ZX 3. Level of Confidence (1 - ) Affects Z © 1984-1994 T/Maker Co. 7 - 21 7 - 22 Intervals & Confidence Level Sampling Distribution of Mean Intervals extend from X - ZX to X + ZX Estimation Example Mean ( Known) The mean of a random sample of n = 25 isX = 50. Set up a 95% confidence interval estimate for if = 10. Assume the population is normally distributed. 100(1 - ) % of intervals contain . 100 % do Large number of intervals not. 7 - 23 7 - 24 Estimation Example Solution Confidence Interval for Population Mean ( Unknown) 1. Assumptions Population Standard Deviation Is Unknown Population is Normally Distributed 2. Use Student’s t Distribution 3. Confidence Interval Estimate We are 95% confident that the population mean lies between 46.08 and 53.92 7 - 25 7 - 26 Degrees of Freedom (df) T statistic If we are sampling from a normal distribution, the t statistic has a sampling distribution very much like that of the z statistic: moundshaped, symmetric, with mean 0. The main difference between the sampling distributions of t and z is that the t statistic is more variable than the z. 7 - 27 1. Number of Values that Are Free to Vary when Calculating a Sample Statistic. 2. Degrees of freedom of the t statistic (for one-sample case) is n1. 3. Example 7 - 28 Degrees of Freedom (df) Example 7 - 29 Degrees of Freedom (df) The smaller the number of degrees of freedom associated with the t statistic, the more spread out the sampling distribution of T will be. 7 - 30 Student’s t Table (Table 2 in Appendix of textbook) Student’s t Distribution /2 Standard Normal Bell-Shaped t (df = 13) Assume: n=3 df = n - 1 = 2 = .10 /2 =.05 Symmetric t (df = 5) ‘Fatter’ Tails .05 Z t 0 t values 7 - 32 7 - 31 Estimation Example Mean ( Unknown) Student’s t Table /2 Assume: n=3 df = n - 1 = 2 = .10 /2 =.05 A random sample of n = 25 hasx = 50 & s = 8. Set up a 95% confidence interval estimate for . Assume normal population. .05 t values 2.920 7 - 33 7 - 34 Estimation Example Solution Confidence Interval Estimates Confidence Intervals Mean Known 7 - 35 7 - 36 Proportion Unknown Confidence Interval Proportion Estimation Example Proportion 1. Assumptions A random sample of 400 graduates showed 32 went to grad school. Set up a 95% confidence interval estimate for p. Two Categorical Outcomes Population Follows Binomial Distribution Normal Approximation To The Sampling Distribution Of Can Be Used and 2. Confidence Interval Estimate 7 - 37 7 - 38 Estimation Example Solution Precision of estimate • • • The smaller the width of CI, the higher the precision of estimation. What are the ways to make the estimate more precise? In practice, if we choose to increase sample size to increase precision, we should also take into account the costs of study. 7 - 39 7 - 40 CI for Finite populations In cases where n/N > 0.05, the formulas for CI must be adjusted (as the standard error of estimate changes) Please consult your textbook or reference books if you are interested in the adjusted formulas. Knowing these formulas is not required for the test/exam. 7 - 41 Conclusion 1. State What Is Estimated 2. Distinguish Point & Interval Estimates 3. Explain Interval Estimates 4. Compute Confidence Interval Estimates for Population Mean & Proportion 7 - 42