1 Econ 310 Professor Wallace October 17, 2013 Lecture: - The first midterm exam - Estimation (Chapter 10, 12) o Point Estimation o Interval Estimation Midterm #1 Descriptives - Each TA graded a different question for consistency. If you have concerns about how a question was graded you should contact the TA that did the graded via email. o Zack graded #1 ([email protected]) o Xianwei graded #2 ([email protected]) o Pedro graded #3 ([email protected]) o Zhewen graded #4 ([email protected]) o Leo (Chenxin) graded in #5 ([email protected]) - The mean was 52 points (out of 120) o Question 1 mean was 21 o Question 2 mean was 5 (ouch~) o Question 3 mean was 9 o Question 4 mean was 10 o Question 5 mean was 7 - The mean was also 52 points. - The high score was 101.5. There were 3 scores at 100 or above - The low score was below 15.2 - Distribution of scores - Midterm #1 Curve - Considerations o The exam was too long to complete, but I’ve given a lot of thought to it and I don’t think the questions were too difficult. o There is a tradeoff because it is difficult to write a exam that spans the scope of topics that we covered that is short that it can be completed in 75 minutes. o Because of the length the exam was basically a 100 point exam rather than a 120 point exam. A 80+ AB 65-79.5 B 55-64.5 BC 45-54.5 C 35-44.5 D 25-34.5 F <25 05101520Percent0 10 20 30 40 50 60 70 80 90 100 110 120Midterm 1 Scores (out of 120)3 - Distribution of Grades 05101520PercentF D C BC B AB AGrade4 Point Estimation Point estimator – draws inference about a population by estimating the value of an unknown parameter using a single point value. Sampling error – the difference between the sample statistic and the population parameter Illustration: When using the sample mean to estimate the population mean the sampling error is X Standard error - The standard error is simply the standard deviation of the sampling error. Illustration: When using the sample mean to estimate the standard error is XXSVar Xnn or in the special case where the population is binomial and X is a sample proportion P 11PPppppnn 5 Properties of point estimators - Unbiased – an unbiased estimator of a population parameter is an estimator with an expected value equal to the parameter. Illustration: Assuming that iX is independently distributed with mean and variance 2 the sample mean X is an unbiased estimator for because EX. Thus, the expected value of the sampling error is equal to zero. - Consistency – an unbiased estimator is said to be consistent if the difference between the estimator and the parameter grows smaller as the sample size grows larger. Illustration: The sample mean X is a consistent estimator for because lim 0nPX This result should make sense because X is unbiased and lim lim0nnVar X Var X or, in other words, the variance difference between estimator and the population goes to zero as n gets large. - Relative efficiency – if there are two unbiased estimators of a parameter, the one whose variance is smaller is said to have relative efficiency6 Interval Estimation Interval estimator – draws inference about a population parameter by estimating the value using an interval Example 1: Assume that we have taken a sample of size 10,000 and compute 30,000x and 20,000. Find an interval such that we are 95% certain that will lie in the interval. For any standard normal random variable we know that 1.96 1.96 0.95PZ If this is true for any standard normal random variable then it should be true for Xn. Thus, 1.96 1.96 0.95XPn What we want to do is solve the inequality inside for 7 Multiply through by n 1.96 1.96 0.95PXnn Multiply through by -1 and switch inequality signs. 1.96 1.96 0.95PXnn Rearrange so that smaller bound is on the left 1.96 1.96 0.95PXnn Add X to both sides 1.96 1.96 0.95P X Xnn What the above statement says is that 95% percent of the time we will draw a sample mean that is within 1.96 standard deviations of .8 1.96 1.96 0.95P X Xnn Implication: The interval 1.96xn will contain for 95% of the values of X - The 95% of the time that we draw a value of X within one 1.96 standard deviations of the (its mean) the interval will include - The 5% of the time that we draw a value of X farther than 1.96 standard deviations from (its mean) our confidence the interval will not include - We can never be completely certain whether the interval contains or not9 For our specific example we can be 95% confident that that the population mean is in the interval 20,00030,000 1.9610,000 or between 29,608 and 30,392 In this example: - 95% is the confidence level - The confidence coefficient is 0.95 - The significance level ( ) = 1 – Confidence Coefficient General formula for 1 100% confidence interval: /2SxZn where: - S is the sample standard deviation (use the population standard deviation if you have it); - 1 is the confidence coefficient; and - /2Z is the Z value providing an area /2in the upper tail of a standard normal probability distribution. Example 2: Using the same numbers calculate 90% confidence interval. We just use different Z values in your calculations. Here we want 0.05Z instead of 0.025Z 90% confidence interval for : 20,00030,000 1.64510,000 30,000 1.645 200 30,000 329 We can be 90% confident that the population mean is between 29,671 and
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