Download Inferential Statistics: Estimation and Confidence Intervals (ECON 413, Fall 2004) and more Exams Economics in PDF only on Docsity! ECON 413 Fall 2004 Estimation Inferential statistics is concerned with making inferences about a population using the information contained in a sample. Inferential statistics involves estimating unknown parameters, testing hypotheses about them, and forecasting the future values of variables of interest. In estimation, we are interested in estimating the characteristics of the population parameters. The mean of a population µ, the population proportion p, and the differences between the means of two proportions µ1-µ2 are examples of parameters. For example, we may wish to estimate mean SAT mathematics score for the more than 250.000 high school seniors in California. Unless we give the test to all the students, there is no way of knowing the mean SAT-M score because only 45% of the students in California take the test and it is likely that the students who take the test are different in characteristics than the students who don’t take the test. However, we can give the test to a simple random sample (SRS) from the population of seniors and use the sample average to estimate the unknown parameter. The sample average would be a desirable estimator because we know from our earlier discussion that it is an unbiased estimator of the mean and furthermore the law of large numbers tells us that the sample mean approaches the mean as the sample size becomes large. We can estimate a parameter either using a point estimate or a confidence interval. For example, if we give the SAT to a SRS of 500 students from the population of seniors and if the sample average score turns out to be 461, we can use this number as an estimate of the population mean score. The number 461 is a point estimate. Its major shortcoming is that it does not provide any information about how accurate this number is as an estimate of the mean. For this reason, we prefer to use interval estimates known as confidence intervals. Confidence intervals are constructed using information about the sampling distributions of sample statistics. For example, we can construct a confidence interval for µ by using the information that the sampling distribution of x is normal (or approximately normal) with mean µ and standard deviation /x nσ σ= . To give a specific example, let’s suppose that σ =100. This means that 100 / 500xσ = = 4.5. Then, we know that the probability that x lies between 2 standard deviations of the unknown mean is about 95%, i.e. ( 2 4.5 2 4.5) 0.95P xµ µ− × < < + × = But, note that, this also means that the probability that the mean µ lies within 2 standard deviations of x is also 0.95, i.e. ( 2 4.5 2 4.5) 0.95P x xµ− × < < + × =
