Paired sample t test
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Paired sample t&-test (t&-test for related data)
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We have been considering cases where we wish to test whether two sample means are significantly different. Up to now we have considered cases where the two samples are independent, random samples &- there is no association between the samples. However, we shall now consider the case where one value in one of the samples is related to another value in the second sample. The values are paired and the size of each sample is the same. We also are concerned with the case where the common sample variance is not known and has to be estimated. In this case the test is to apply is the t&-test for related samples. Let (X1, X2) represent a pair of values where X1 is drawn from the first sample and X2 is drawn from the second sample. Then the difference for each pair is D= X1&- X2. [Equation goes here &- download the original to see it.] We assume that the mean of the difference is normally distributed [Equation goes here &- download the original to see it.] where [Equation goes here &- download the original to see it.] is the unbiased estimate of the variance of the difference and n is the common sample size. We estimate [Equation goes here &- download the original to see it.] in the usual way [Equation goes here &- download the original to see it.] Then the appropriate test statistic is&: [Equation goes here &- download the original to see it.] The critical value is drawn from the Student's t&-distribution with degrees of freedom v = n&-1. We are in a position to immediately illustrate this result. Example It is believed that airfares between London and Budapest have fallen between 1997 and 1998. Travel agencies were sampled at random and their cheapest return air fare between these two celebrated cities was noted for the first week of January 1997 and 1998. The results were&: [Table goes here &- download the original to see it.] By means of a paired&-sample t&-test at the 5% significance level determine whether the belief that airfares have fallen is valid. Let the 1997 fare be X and the 1998 fare be Y. Let [Equation goes here &- download the original to see it.] be the main air fares in 1997 and1998 respectively. H0&: [Equation goes here &- download the original to see it.] H1&: [Equation goes here &- download the original to see it.] one&-tailed [Equation goes here &- download the original to see it.] Let D = X-Y, then we test these hypotheses by their equivalents. [Equation goes here &- download the original to see it.] We need to estimate from the data and estimate the variance of the differences. [Table goes here &- download the original to see it.] [Equation goes here &- download the original to see it.]
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Contents of
Paired sample t test
1 Paired sample t&-test (t&-test for related data)
2 One&-step formula for paired sample t&-test (t&-test for related data)
3 Example of paired sample t&-test (t&-test for related data)
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