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Unformatted text preview: Created by Professor McGuckian 1 Inferences Based on Two Samples In the following sections, our goal is to compare two population parameters to each other. We want to know the relationship between the parameters (if they are equal or if one is larger than the other). For example, I may want to compare the entrance exam results for men and women trying to be admitted to a prestigious prep school. In that scenario, we would look at the mean scores men and women and see if there is a difference between them. Example: Since court liability awards vary over time, an insurance company wants to compare the mean level of current personal liability awards with those from one year earlier. Random samples of cases were selected from each year. The data is summarized below: Year Sample Size Sample Mean Sample Variance Current 50 1.32 0.9734 Previous 55 1.04 0.7291 If we want to estimate the true difference between the average amounts of awards over the two years , our best point-estimate of that difference is 1 2 X X- the difference of the sample means. If we also would like to form a confidence interval using the same format as we used in earlier sections, we need to know some properties of the sampling distribution of 1 2 X X- . To understand these properties note that ( ) X Y E X Y = and ( ) ( ) 2 2 2 , X Y Var X Y Cov X Y = + . Properties of the Sampling Distribution of 1 2 X X- : 1. 1 2 1 2 X X - =- 2. 1 2 2 2 1 2 1 2 X X n n - = + 3. If the sampled populations are normally distributed then so is the distribution of 1 2 X X- regardless of the sample size. 4. If the sampled populations are not normal then we will need to have large sample sizes to ensure that we can approximate the distribution of 1 2 X X- by the normal distribution. ***Note: We are assuming that the samples drawn are independent. Note: If 2 2 1 2 &amp; are unknown, we may use their sample estimates as approximations as long as the sample sizes are large (&gt; 30). Created by Professor McGuckian 2 Using the properties above and the same structure as we used in previous sections, we can find a formula for the: (1- )100% Confidence Interval for the True Difference Between the Population Means Remember, that if we do not know the population standard deviation, but the sample size is large we can use the sample estimates. In other words, 1 2 2 2 1 2 1 2 / 2 ( ) 1 2 /2 1 2 2 2 1 2 1 2 /2 1 2 ( ) ( ) ( ) x x x x z x x z n n s s x x z n n -- =- + - + Now lets use the above formula we just found on the opening example regarding court awards: Steps for constructing the Confidence Interval for the True Difference between the Population Means: Step 1 Gather Data from Problem, Calculate 1 2 X X- , and Calculate 2 2 1 2 1 2 n n + ....
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- Fall '08