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Handout_11 CONFIDENCE INTERVALS FOR DIFFERENCE BETWEEN TWO SAMPLE MEANS

# Handout_11 CONFIDENCE INTERVALS FOR DIFFERENCE BETWEEN TWO SAMPLE MEANS

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May 17’09 SAMPLING DISTRIBUTION OF THE DIFFERENCE #11 BETWEEN TWO SAMPLE MEANS. CONSTRUCTION OF THE CONFIDENCE INTERVAL Frequently the interest in a statistical investigation is focused on two populations. For example, an investigator may wish to know something about the difference between two population means to find out, if it is reasonable to conclude that two population means are different. If it was found that the population means are different, then the investigator may wish to know by how much they differ. Thus, the magnitude of the difference between two population means may be the subject of interest. In investigations of this type knowledge of the properties of sampling distribution of the difference between two means is important. Example . Suppose we have two populations ( 1 and 2 ) of individuals – the population 1 has experienced some condition thought to be associated with mental retardation, and the other population 2 has not experienced the condition. The distribution of intelligence scores in each of the two populations is believed to be approximately normal with a standard deviation of 20 . Suppose, further, that we take a sample of 15 individuals from each population and compute for each sample the mean intelligence score with the following results: _ _ x 1 * = x mean_1 * = 92 and x 2 * = x mean_2 * = 105 . If there is no difference between the two populations, with respect to their true mean intelligence scores, what is a probability of observing a difference between two sample means that is less than or equal to the obtained value of (x mean_1 * – x mean_2 * ) = –13 ? Solution . To answer this question we need to know the nature of the sampling distribution of the relevant statistic, the difference between two sample means , x mean_1 – x mean_2 . Although, in practice, we would not attempt to construct the desired sampling distribution, we can conceptualize the manner in which it could be done when sampling is from finite populations. We would begin by selecting from population 1 (of size N 1 ) all possible samples of size n 1 = 15 and computing the mean for each sample. Similarly, we would select from population 2 (of size N 2 ) all possible samples of size n 2 = 15 and compute the mean for each of these samples. We would then take all possible

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Handout_11 CONFIDENCE INTERVALS FOR DIFFERENCE BETWEEN TWO SAMPLE MEANS

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