1/08/03 252oneal (Open this document in 'Outline' view!)A. Parameter Estimation1. Review of the Normal Distribution2. Point and Interval Estimation3. A Confidence Interval for the Mean when thePopulation Variance is Known.a. A Two-Sided Confidence Intervalb. A One-Sided Confidence Interval.4. A Confidence Interval for the Mean when thePopulation Variance is not Known.5. Deciding on Sample Size when working with a Mean6. A Confidence Interval for a Proportion.(a. Small Samples.b. Large Samples.c. Deciding on Sample Size.7. A Confidence Interval for a Variance.(8. AppendixA Confidence Interval for a Median.1/08/03 252oneal (Open this document in 'Outline' view!) ECONOMICS 252 COURSE OUTLINEA. Parameter Estimation1. Review of the Normal DistributionSee 251greatD, 251distrex2, 251distrex3, 251distrex42. Point and Interval Estimation3. A Confidence Interval for the Mean when the Population Variance is Known.a. A Two-Sided Confidence Interval An interval of this type is used in two situations: (i) where the population variance, 2, is in fact, known and the sample size is relatively large; or (ii) where the variance is not known and the sample variance, 2s, is used to replace 2, but the degrees of freedom are so large that the appropriate value of 1nt is not very different from z. The first of these situations is not very realistic, but serves as a good introduction to confidence intervals. The formula for this type of confidence interval for the mean is, xzx2, where nx. Note: If ,05. Nn use 1NnNnx(nis sample size and Nis population size) See 252onealex1.Don’t use this method unless you know the population variance.b. A One-Sided Confidence Interval.There are two types of one-sided confidence interval for the mean.These are (i) An upper bound, and (ii) a lower bound, and have the form: xzx and xzx. An example is in 252oneaex1a.4. A Confidence Interval for the Mean when the Population Variance is not Known."The variance is not known " implies that there is no previous knowledge or assumption about the value of 2. Knowing 2s is having a guess as to what the variance is; it is not the same as knowing 1the variance. If the population distribution is normal or approximately normal, the formula for a two-sided confidence interval for the mean is xnstx12, where nssx. Note: If ,05. Nn use 1NnNnssx See 252onealex2 and 252oneaex3.Note: this is the more common case – if you do not know the population variance and the sample size is not very large, using z instead of t is a very bad idea.25. Deciding on Sample Size when working with a MeanThe formula usually suggested is 222ezn, where, if is not known, it can be approximated by 6999.001.xx .6. A Confidence Interval for a Proportion.(a. Small Samples.Table 16 (ConfidenceIntervalsBinominalDistribution.pdf) gives Confidence Intervals for proportions. These tables are of use when the conditions do not exist in which one can use the normal distribution. For example if 5.p and 10 n, and we wish to find a 95% confidence interval, we can look at the horizontal axis of the upper table.There we can find 5.p and look up to find the upper and lower curves for 10n. Then vertical line at 5.p intersects these curves. The lower curve meets the vertical line at about 175.p. (Read up the vertical axis.). The upper curve meets the vertical line at about 825.p, so that our 95% confidence interval is about 825.175. p.) b. Large Samples.More usually, using the normal approximation to the binomial distribution, and using p for the population probability of success and q for the population probability of failure, and letting pand q be the corresponding sample quantities, we can write pszpp2, where nqpspand pq 1. An example is in 251 proport.c. Deciding on Sample Size.The usually suggested formula is 22epqzn , but since p is usually unknown, a conservative choice is to set 5.0p. This is the formula everyone forgets that we covered.7. A Confidence Interval for a Variance.This method is only appropriate when the population distribution is normal or approximately normal.For small samples 2122222211 snsn, 3but if the degrees of freedom are too large for the chi-square table use DFzDFsDFzDFs222222. An example is in 252oneaex4.4(8. AppendixA Confidence Interval for a Median.In a situation where the population distribution is not normal,it is often more appropriate to find the median than the mean. The process of finding a confidence interval for a median is based on one simple fact: the probability that a single number picked at random from a population is above (or below) the median is 50%. Similarly, the probability that any two numbers picked at random from a population are both above (or both below) the median is 25%.. This comes from themultiplication rule: If A is the probability that the first number is abovethe median, and B is the probability that the second number is above the median, then and if BABPAPBAP are independent events. If the probability of both numbers being above the median is 25%, and the probability of both numbers being below the median is 25%, then the probability that both numbers are on the same side of the median is 50%. This is due to the addition rule: Let event C be "both numbers are above the median," and event D be "both numbers are below the median." Then event DC is "both numbers are on the same side of the median." The addition rule says that if DC and are mutually exclusive, )()( DPCPDCP . Finally, if the probabilitythat both numbers are on the same side of the median is 50%, then the probability that the two numbers are on opposite sides of the median is also 50%. This means that, since any two numbers picked from the sample have a 50% chance of bracketing the median, these two numbers constitute a 50% confidence interval.Note that, since p, the probability that any one number is above the median, is 0.5, and q, the probability that any one number isbelow the median, is also 0.5, we have a problem that resembles finding the distribution of the number of heads on two tosses of a fair coin. If we call a head a success, the distribution of heads on two tosses
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