1. Ka-fu Wong © 2003 Chap 8- 1 Dr. Ka-fu Wong ECON1003 Analysis of Economic Data

2. Ka-fu Wong © 2003 Chap 8- 2 l GOALS Explain the central limit theorem and confidence interval together from a different perspective. Additional materials The Central Limit Theorem and Confidence Interval

3. Ka-fu Wong © 2003 Chap 8- 3 Empirical histogram #1 Two balls in the bag: Draw 1 ball 1000 times with replacement. Plot a relative frequency histogram (empirical probability histogram). 0.5 The empirical histogram looks like the population distribution !!! What is the probability of getting a red ball in any single draw? 0.5

4. Ka-fu Wong © 2003 Chap 8- 4 Empirical histogram #2 5 balls in the bag: Draw 1 ball 1000 times with replacement. Plot a relative frequency histogram (empirical probability histogram). 0.6 The empirical histogram looks like the population distribution !!! What is the probability of getting a red ball in any single draw? 0.6 0.4

5. Ka-fu Wong © 2003 Chap 8- 5 Empirical histogram #3 5 balls in the bag: Draw 1 ball 1000 times with replacement. Plot a relative frequency histogram (empirical probability histogram). 0 0.2 The empirical histogram looks like the population distribution !!! What is the probability of getting a “three” in any single draw? 0.2 0 1 2 3 4 1 2 3 4 What is the expected value (i.e., population mean) of a single draw? 0.2*0 + 0.2*1 + … + 0.2*4 = 2 Variance = 0.2*(-2) 2 + 0.2*(-1) 2 +… +0.2*(2) 2 = 2

6. Ka-fu Wong © 2003 Chap 8- 6 Empirical histogram #3 continued MeansCombinations 0.00,0 0.50,1 1,0 1.01,1 2,0 0,2 1.51,2 2,1 3,0 0,3 2.01,3 3,1 2,2 4,0 0,4 2.51,4 4,1 3,2 2,3 3.03,3 4,2 2,4 3.53,4 4,3 4.04,4 5 balls in the bag: Draw 2 balls 1000 times with replacement. Compute the sample mean. Plot a relative frequency histogram (empirical probability histogram) of the 1000 sample means. 0 1 2 3 4 All combinations are equally likely.

7. Ka-fu Wong © 2003 Chap 8- 7 Empirical histogram #3 continued 5 balls in the bag: Draw 2 ball 1000 times with replacement. Compute the sample mean. Plot a relative frequency histogram (empirical probability histogram) of the 1000 sample means. 0 1 2 3 4 What is the probability of getting a sample mean of 2.5 in any single draw? 0.16 What is the expected sample mean of a single draw? 0.04*0 + 0.08*0.5 +… + 0.04*4 = 2 Variance of sample mean = 0.04*(-2) 2 + 0.08 *(-1.5) 2 + … + 0.04*(2) 2 = 1

8. Ka-fu Wong © 2003 Chap 8- 8 Central Limit Theorem #1 5 balls in the bag: Draw n (n>30) ball 1000 times with replacement. Compute the sample mean. Plot a relative frequency histogram (empirical probability histogram) of the 1000 sample means. 0 1 2 3 4 The Central Limit Theorem says 1.The empirical histogram looks like a normal density. 2.Expected value (mean of the normal distribution) = mean of the original population mean = 2. 3.Variance of the sample means = variance of the original population /n = 2/n.

9. Ka-fu Wong © 2003 Chap 8- 9 Central Limit Theorem #2 Some unknown number of numbered balls in the bag: We know only that the population mean is  and the variance is  2. Draw n (n>30) ball 1000 times with replacement. Compute the sample mean. Plot a relative frequency histogram (empirical probability histogram) of the 1000 sample means. 0 1 2 3 4 The Central Limit Theorem says 1.The empirical histogram looks like a normal density. 2.Expected value (mean of the normal distribution) = . 3.Variance of the sample means =  2 /n. ? ?

10. Ka-fu Wong © 2003 Chap 8- 10 Confidence interval #1 Some unknown number of numbered balls in the bag: We know only that the population mean is  and the variance is  2. 0 1 2 3 4 The Central Limit Theorem says 1.The empirical histogram looks like a normal density. 2.Expected value (mean of the normal distribution) = . 3.Variance of the sample means =  2 /n. ? ? What is the probability that the sample mean of a randomly drawn sample lies between    /  n ? 0.6826

11. Ka-fu Wong © 2003 Chap 8- 11 Confidence interval #2 Some unknown number of numbered balls in the bag: We know only that the population mean is  and the variance is  2. 0 1 2 3 4 The Central Limit Theorem says 1.The empirical histogram looks like a normal density. 2.Expected value (mean of the normal distribution) = . 3.Variance of the sample means =  2 /n. ? ? What is the probability that the “unknown” population mean (  ) lies between an interval around the mean of a randomly drawn sample ? x   /  n ? 0.6826 X +  /  n -  /  n

12. Ka-fu Wong © 2003 Chap 8- 12 - END - Additional materials The Central Limit Theorem and Confidence Interval