Difference of two sample means
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Difference of two sample means&: The z&-test
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Suppose we have two independent samples drawn one from each of two normally distributed populations of differing known variance. That is, we are given that [Equation goes here &- download the original to see it.] where [Equation goes here &- download the original to see it.] are known. We are asked to test whether the two population means are equal at a given significance level. To say that the samples are independent is to say that values of X are not in any way associated or paired with values of X2. [Equation goes here &- download the original to see it.] To carry out this test we test the difference of the means, for it turns out that [Equation goes here &- download the original to see it.] is normally distributed if both X1 and X2 are normally distributed. [Equation goes here &- download the original to see it.] Hence the test statistic is&: [Equation goes here &- download the original to see it.] The critical value is found using the tables for the standardised normal distribution and finding the z value corresponding to the given significance level. The first example illustrates the case where the population variances are known and different. Example (1) A random sample is taken from each of two normally distributed populations. The sample size of the first population is 10 with sample mean 18.2. The sample size of the second population is 12, with sample mean 15.3. It is known that the population variances are 2.5 and 2.8 respectively. Test at the 1% level whether the means of the two populations are equal. First sample&: [Equation goes here &- download the original to see it.] [Equation goes here &- download the original to see it.] Second sample [Equation goes here &- download the original to see it.] [Equations go here &- download the original to see it.] When it is given that the population variances are identical, then the test becomes even simpler, since then [Equation goes here &- download the original to see it.]
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Contents of
Difference of two sample means
1 Difference of two sample means&: The z&-test
2 Example of difference of two sample means&: The z&-test
3 The t&-test for unrelated data
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