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Confidence Interval Estimation Dr. Geoffrey M. Ducanes UP School of Economics Session Outline 1. Confidence Intervals for the Mean a. Large Samples b. Small Samples 2. Confidence Intervals for Proportion 3. Confidence Intervals for the Standard Deviation Introduction • This morning you learned about point estimators. • Assuming representative sample • Sample mean is point estimator for population mean. • Sample proportion is point estimator for population proportion. • Sample standard deviation is point estimator for population standard deviation. • Problem with using point estimate is that it is almost never equal to the exact parameter of the population. • [Illustration using Excel] Introduction • Might be better to specify an interval of values and a statement of how confident you are that the interval contains the population parameter. • Confidence Interval: Point estimate ± Margin of Error How to estimate a confidence interval? 1. Compute the point estimate 2. Compute the margin of error 3. Construct the interval endpoints using the point estimate and the margin of error Confidence Interval for the Mean (large sample or known) 1. Find the sample statistics n and 𝑥. 2. Specify , if known. Otherwise if n 30, use the sample standard deviation s as estimate for . 3. Find the critical value 𝑧𝑐 that corresponds to the given level of significance. 4. Find the margin of error E. 5. Find the left and right endpoints and form the confidence interval. 𝑥 𝑥= 𝑛 𝑠= 𝑥−𝑥 𝑛−1 2 Use the standard Normal Table or Excel. 𝐸 = 𝑧𝑐 𝜎 𝑛 Interval: 𝑥 - E < < 𝑥 + E Question • From the formula, why is the effect of a larger sample size on the width of the interval? Example • Ilocos Norte HH income and expenditure (CI data – EPDP Training) Source: Caldwell, S. 2010. Statistics Unplugged Confidence Interval for the Mean (small sample, unknown) • • In many situations, population standard deviation is unknown and large sample is not available. Follow same procedure but use t-distribution Confidence Interval for the Mean (small sample, unknown) 1. Find the sample statistics n, 𝑥, and s. 2. Identify the degrees of freedom, the level of confidence c, and the critical value 𝑡𝑐 . 3. Find the margin of error E. 4. Find the left and right endpoints and form the confidence interval. 𝑥 𝑥= 𝑛 𝑠= 𝑥−𝑥 𝑛−1 2 d.f. = n1 Use the t Table or Excel. 𝐸 = 𝑡𝑐 𝑠 𝑛 Interval: 𝑥 - E < < 𝑥 + E Example • Batanes food expenditure and housing and utilities expenditure (CI data – EPDP Training) Confidence Interval for a Population Proportion p • • In cases where binomial distribution can be approximated by a normal distribution (np 5 and nq 5), Construction of confidence interval for p similar to constructing confidence interval for a population mean. Point estimate ± Margin of error 𝑝±E • E=𝑧𝑐 • • 𝑝𝑞 𝑛 Confidence Interval for the Population Proportion p 1. 2. 3. 4. 5. 6. Identify the sample statistics x and n, where x is the number of success. Find the point estimate 𝑝. Verify that the sampling distribution of 𝑝 can be approximated by a normal distribution Find the critical value 𝑧𝑐 that corresponds to the given level of confidence c. Find the margin of error E. Find the left and right endpoints and form the confidence interval. n𝑝 5, n𝑞 5 Use the standard Normal Table or Excel. 𝐸 = 𝑧𝑐 𝑝𝑞 𝑛 Interval: 𝑝- E < < 𝑝+ E Example • Metro Manila households with at least one aircon, at least one car (CI data – EPDP Training) Confidence Interval for the population standard deviation • • • Uses Chi-Square Distribution Confidence Interval for (𝑛−1)𝑠 2 2 𝜒𝑅 << (𝑛−1)𝑠 2 𝜒𝐿2 Confidence Interval for the population standard deviation 1. Verify that the population has a normal distribution. 2. Identify the sample statistic n and the degrees of freedom. 3. Find the point estimate for the population variance 𝑠 2 . 4. Find the critical values 𝜒𝑅2 and 𝜒𝐿2 that correspond to the given level of confidence c. 5. Compute the left and right endpoints to form the confidence interval. d.f. = n -1 𝑠= 𝑥−𝑥 𝑛−1 2 Use Chi-Square Table or Excel. (𝑛−1)𝑠 2 2 𝜒𝑅 << (𝑛−1)𝑠 2 𝜒𝐿2 Example • Ilocos Norte HH income and expenditure (CI data – EPDP Training) References • Larson, R. and B. Farber. 2012. Elementary Statistics: Picturing the World (5th Edition). Prentice Hall • Caldwell, S. 2010. Statistics Unplugged (3rd Edition). Wadsworth CENGAGE Learning. • Danao, R. 2013. Introduction to Statistics and Econometrics. Diliman: UP Press