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Unformatted text preview: Stat 0302B Business Statistics Spring 2010-2011 Chapter V Confidence Intervals and Sample Size Determination § 5.1 Estimation about mean of normal population Suppose the target population is assumed to be normal. We are interested in the unknown parameter : population mean μ . Since it is unknown, we would like to make some inference about it by a random sample. Point Estimation Using a single value calculated from the sample to estimate the interested population parameter is called point estimation . A reasonable point estimate of μ is the sample mean X . On the other hand, we can use the sample standard deviation S to estimate the population standard deviation σ . Example 5.1 A marketing research organization was hired to estimate the mean prime lending rate for banks located in the western region of the United States. A random sample of banks was selected from within the region and has a mean prime rate 50 = n % 1 . 8 = X and standard deviation % 24 . = S . Then we may estimate the values of μ and σ by 8.1% and 0.24% respectively. Interval Estimation We can seldom estimate correctly. It is almost impossible to have a sample mean exactly equal to the population mean. Then, how can we justify our estimate? Can we know how accurate the estimate is? What is our confidence of the closeness? The answer is: we can specify a “ margin of error ” for the point estimate from the sample, which, when added to and subtracted from the sample result, gives us a range of values. We use the whole range to estimate the population mean. Then we can access the quality of this range by considering our confidence that this range contains the actual population mean. Such a range is called a confidence interval. In Example 5.1, the standard error of the sample mean is 034 . 50 / 24 . ≈ . Since there is 95.44% chance that the sample mean is within 2 standard errors from the population mean, we may have high confidence that the population mean prime rate is between % 032 . 8 % 034 . * 2 % 1 . 8 = − and % 168 . 8 % 034 . * 2 % 1 . 8 = + . P.98 Stat 0302B Business Statistics Spring 2010-2011 The value is the margin of error of the point estimate 8.1%. The range 8.032% to 8.168% is the confidence interval for % 068 . % 034 . * 2 = μ . The general form of the margin of error for estimation about the mean (or proportion) is margin of error = critical value × standard error The confidence interval would take the following form Point estimate ± critical value × standard error The critical value, which can be obtained from statistical tables (e.g. standard normal distribution table), depends on the sampling distribution of the point estimate and the required confidence. For a higher confidence level, we may need a larger critical value and it would produce a wider confidence interval....
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- Spring '11