# The sampling distribution of the difference in sample

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The sampling distribution of the difference in sample means is normal . Raising Marks, Raising Money, Raising Roofs
Guelph SOS: Students Offering Support Hypothesis Testing We often use the null hypothesis that there is no difference in population means. There are three possible alternative hypotheses to this: Population mean 1 is greater than Population mean 2 Population mean 1 is smaller than Population mean 2 The Population means are not equal. What we chose depends on what we are trying to test. E.g. if we want to test that a new drug is better , we may use a one- sided hypothesis. If the standard deviation of the populations are known, use a Z test statistic. If the null hypothesis is true, then the statistic will have the standard normal distribution. If the standard deviation of the populations are not known, we have two options. Pooled-variance two-sample t procedure Assumptions Assumes that the population standard deviations are equal to each other. Ratio of differences is what is important. Assumes normally distributed populations. Assumes independent random samples. Advantage : Test statistic is exactly a t-distribution Pooled sample variance Combine sample variances together to get the best estimation of the common population variance. Standard error of the difference in sample means The square root of its estimated variance. Estimated standard deviation of the sampling distribution of the difference in the sample means. p-values Alternative hypothesis: population mean 1 is greater than population mean 2 p-value = area to the right of the test statistic Alternative hypothesis: population mean 2 is greater than population mean 1 p-value = are to the left of the test statistic Alternative hypothesis: population means are not equal p-value = Double the area to the left or right (whichever is small) Degrees of Freedom = the degrees of freedom for the variance estimator Lose two degrees of freedom when we use the two sample means to estimate the two population means. n1 + n2 - 2 When population sizes differ, it is a problem for this procedure. Raising Marks, Raising Money, Raising Roofs
Guelph SOS: Students Offering Support Worst when the small population has the large sample variance. Welch procedure Assumes normally distributed populations. Used when population variances are not equal. The test statistic is similar to that of the pooled-variance t test. The only difference is the form of the standard error in the denominator Standard error of the difference in sample means estimates the true standard deviation of the sampling distribution of the difference between the two sample means. Used in the same kind of situations as pooled-variance t test.