e-book Hypothesis Test Using Paired Observations

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A paired t-test is used to compare two population means where you have two which observations in one sample can be paired with observations in the other.
Table of contents

Paired Samples T-test in R. Preleminary test to check paired t-test assumptions Assumption 1: Are the two samples paired? Yes, since the data have been collected from measuring twice the weight of the same mice. Assumption 2: Is this a large sample? How to check the normality?


What is paired samples t-test?

Compute paired samples t-test Question : Is there any significant changes in the weights of mice after treatment? Note that: if you want to test whether the average weight before treatment is less than the average weight after treatment, type this: t. Interpretation of the result The p-value of the test is 6.

Access to the values returned by t. Online paired t-test calculator You can perform paired-samples t-test , online , without any installation by clicking the following link: Online paired-samples t-test calculator. See also Paired Samples Wilcoxon Test non-parametric.

John H. McDonald

Infos This analysis has been performed using R software ver. Enjoyed this article? Show me some love with the like buttons below Thank you and please don't forget to share and comment below!! Montrez-moi un peu d'amour avec les like ci-dessous We conclude that the mean of variable write is different from N — This is the number of valid i.

Paired samples t-test

Error Mean — This is the estimated standard deviation of the sample mean. If we drew repeated samples of size , we would expect the standard deviation of the sample means to be close to the standard error. The standard deviation of the distribution of sample mean is estimated as the standard deviation of the sample divided by the square root of sample size: 9. It is the ratio of the difference between the sample mean and the given number to the standard error of the mean: Since the standard error of the mean measures the variability of the sample mean, the smaller the standard error of the mean, the more likely that our sample mean is close to the true population mean.

This is illustrated by the following three figures. In all three cases, the difference between the population means is the same. But with large variability of sample means, second graph, two populations overlap a great deal.

An Example of a Paired-Difference t Test and Confidence Interval

Therefore, the difference may well come by chance. On the other hand, with small variability, the difference is more clear as in the third graph.

The smaller the standard error of the mean, the larger the magnitude of the t-value and therefore, the smaller the p-value. We loose one degree of freedom because we have estimated the mean from the sample. We have used some of the information from the data to estimate the mean, therefore it is not available to use for the test and the degrees of freedom accounts for this.

Sig 2-tailed — This is the two-tailed p-value evaluating the null against an alternative that the mean is not equal to It is equal to the probability of observing a greater absolute value of t under the null hypothesis. If the p-value is less than the pre-specified alpha level usually. For example, the p-value is smaller than 0. Mean Difference — This is the difference between the sample mean and the test value.


A confidence interval for the mean specifies a range of values within which the unknown population parameter, in this case the mean, may lie. It is given by.

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In the example below, the same students took both the writing and the reading test. Hence, you would expect there to be a relationship between the scores provided by each student.

Paired Sample T-Test - Statistics Solutions

The paired t-test accounts for this. For each student, we are essentially looking at the differences in the values of the two variables and testing if the mean of these differences is equal to zero. In this example, the t-statistic is 0. The corresponding two-tailed p-value is 0. We conclude that the mean difference of write and read is not different from 0. This value is estimated as the standard deviation of one sample divided by the square root of sample size: 9.

This provides a measure of the variability of the sample mean. Correlation — This is the correlation coefficient of the pair of variables indicated. This is a measure of the strength and direction of the linear relationship between the two variables. Point Estimate: the sample mean difference is the point estimate of. Note that the standard error of is where s d is the standard deviation of the differences. If the absolute value of the calculated t-statistic is larger than the critical value of t, we reject the null hypothesis.

The general form for a confidence interval around a difference in means is. Suppose we wish to determine if the cholesterol levels of the men in Dixon and Massey study changed from to We will use the paired t-test. Since Specifically, there was an average decrease of To perform a paired t-test in SAS, comparing variables X1 and X2 measured on the same people, you can first create the difference as we did above, and perform a one sample t-test of:.