1 08 03 252oneal Open this document in Outline view ECONOMICS 252 COURSE OUTLINE A Parameter Estimation 1 Review of the Normal Distribution See 251greatD 251distrex2 251distrex3 251distrex4 2 Point and Interval Estimation 3 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 s 2 is used to replace 2 but the degrees of freedom are so large that the appropriate value of t n 1 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 x z x where x 2 Note If n 05 N use x n N n N 1 n n is sample size and N is 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 x z x and x z x 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 s 2 is having a guess as to what the variance is it is not the same as knowing 1 the variance If the population distribution is normal or approximately normal the formula for a two sided confidence interval for the mean is x t n 1 s x where s x s 2 Note If n 05 N use s x s n N n N 1 n 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 2 5 Deciding on Sample Size when working with a Mean The formula usually suggested is n it can be approximated by z 2 2 e2 where if is not known x 001 x 999 6 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 n 10 and p 5 and we wish to find a 95 confidence interval we can look at the horizontal axis of the upper table There we can find p 5 and look up to find the upper and lower curves for n 10 Then vertical line at p 5 intersects these curves The lower curve meets the vertical line at about p 175 Read up the vertical axis The upper curve meets the vertical line at about p 825 so that our 95 confidence interval is about 175 p 825 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 p and q be the corresponding sample quantities we can write where s p p p z 2 s p pq and q 1 p An example is in 251 proport n c Deciding on Sample Size pqz 2 but since p e2 is usually unknown a conservative choice is to set p 0 5 This is the formula everyone forgets that we covered The usually suggested formula is n 7 A Confidence Interval for a Variance This method is only appropriate when the population distribution is normal or approximately normal For small samples n 1 s 2 22 2 n 1 s 2 12 2 3 but if the degrees of freedom are too large for the chi square table use s 2 DF z 2 2 DF s 2 DF z 2 2 DF An example is in 252oneaex4 4 8 Appendix A 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 the multiplication rule If A is the probability that the first number is above the median and B is the probability that the second number is above the median then P A B P A P B if A and B 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 C D is both numbers are on the same side of the median The addition rule says that if C and D are mutually exclusive P C D P C P D Finally if the probability that 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 is below 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 is described by the binomial distribution with n the number of tries set at 2 and p the probability of success on one try set at 0 5 For convenience we will use q the probability of failure on one try for the probability that one number is below the median or of getting a tail on one toss of a fair coin It is always true that q 1 p The formula for the binomial distribution is P x C xn p x q n x where x is the number of successes For the probability of two successes heads in 2 tries we find that P 2 C 22 5 2 5 0 1 …
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