Paired Samples Statistics gives univariate descriptive statistics (mean, sample size, standard deviation, and standard error) for each variable entered. Notice that the sample size here is 398; this is because the paired t-test can only use cases that have non-missing values for both variables * A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample*.. This tutorial explains the following: The motivation for performing a paired samples t-test. The formula to perform a paired samples t-test. The assumptions that should be met to perform a paired samples t-test Standard Error (SE) of Paired Mean formula. Sample and Population Statistics formulas list online Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a repeated measures t-test).. A typical example of the repeated measures t-test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure-lowering. Example of paired sample t-test. Let us consider a simple example of what is often termed pre/post data or pretest Р posttest data. Suppose you wish to test the effect of Prozac on the well-being of depressed individuals, using a standardised well-being scale that sums Likert-type items to obtain a score that could range from 0 to 20

- Voorbeeld Paired Samples T-Test, hier vind je hoe je deze test uitvoert in SPSS, hoe deze test nu precies werkt en hoe je de uitkomst moet interpreteren. Indien je daarna vragen hebt staat het team van Afstudeerbegeleider voor je klaar om je persoonlijk te helpen
- Paired Samples T-Test Output. SPSS creates 3 output tables when running the test. The last one -Paired Samples Test- shows the actual test results. SPSS reports the mean and standard deviation of the difference scores for each pair of variables. The mean is the difference between the sample means. It should be close to zero if the populations means are equal
- Two-Sample T-Test from Means and SD's Introduction This procedure computes the two -sample t-test and several other two -sample tests directly from the mean, standard deviation, and sample size. Confidence intervals for the means, mean difference, and standard deviations can also be computed
- Soorten t-testen. Er zijn verschillende soorten t-testen, zoals de one sample t-test, de independent samples t-test en de paired samples t-test.Welke je gebruikt is afhankelijk van wat voor gemiddelden je met elkaar wilt vergelijken. One sample t-test. Gebruik de one sample t-test om te analyseren of het gemiddelde van een steekproef verschilt van een bepaalde waarde

- e more groups or larger sample sizes, there are other tests more accurate than t-tests such as z-test, chi-square test or f-test. Important: The t-test rejects or fails to reject the null hypothesis, never accepts it. 2
- Paired-Samples T Test Data Considerations. Data. For each paired test, specify two quantitative variables (interval level of measurement or ratio level of measurement). For a matched-pairs or case-control study, the response for each test subject and its matched control subject must be in the same case in the data file. Assumptions
- Difference between means of paired samples (paired t test). When the effects of two alternative treatments or experiments are compared, for example in cross over trials, randomised trials in which randomisation is between matched pairs, or matched case control studies (see Chapter 13 ), it is sometimes possible to make comparisons in pairs
- The Paired-Samples T Test procedure compares the means of two variables for a single group. The procedure computes the differences between values of the two variables for each case and tests whether the average differs from 0. The procedure also automates the t-test effect size computation. Exampl
- This video demonstrates how to calculate the effect size (Cohen's d) for a Paired-Samples T Test (Dependent-Samples T Test) using SPSS and Microsoft Excel. C..

* A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample*. This tutorial explains how to conduct a paired samples t-test in Stata. Example: Paired samples t-test in Stata. Researchers want to know if a new fuel treatment leads to a change in the average mpg of a certain car Chapter 13 :: Paired- and Independent-Samples t Tests 319 Figure 13.2. Paired-Samples T Test Output for OKSPEECH and OKTEACH for 1980 GSS Young Adults The Paired Samples Statistics shows the mean for each variable. These means are based on the . 293 persons in the data set with valid scores on both variables. Cases without valid scores on on

Statistics: 1.1 Paired t-tests Rosie Shier. 2004. 1 Introduction A paired t-test is used to compare two population means where you have two samples in which observations in one sample can be paired with observations in the other sample. Examples of where this might occur are The three main types of t-test are independent sample t-test, paired sample t-test, and one sample t-test. An independent samples t-test compares the means for two groups. A paired sample t-test compares means from the same group at different times - one year apart, for example. A one sample t-test tests the mean of a single group against a.

Paired t-test using Stata Introduction. The paired t-test, also referred to as the paired-samples t-test or dependent t-test, is used to determine whether the mean of a dependent variable (e.g., weight, anxiety level, salary, reaction time, etc.) is the same in two related groups (e.g., two groups of participants that are measured at two different time points or who undergo two different. ** One-sample t-test formula**. As mentioned above, one-sample t-test is used to compare the mean of a population to a specified theoretical mean (\(\mu\)). Let X represents a set of values with size n, with mean m and with standard deviation S. The comparison of the observed mean (m) of the population to a theoretical value \(\mu\) is performed with the formula below

Paired vs unpaired t-test. The key differences between a paired and unpaired t-test are summarized below. A paired t-test is designed to compare the means of the same group or item under two separate scenarios. An unpaired t-test compares the means of two independent or unrelated groups 1. To perform a paired t-test, select Tools/ Data Analysis / t-test: Paired two sample for means.. 2. In the t-test: Paired two sample for means dialog box: For the Input Range for Variable 1, highlight the 8 values of Score in group Before (values from 162 to 170). For the input range for Variable 2, highlight the eight values of Score in group After (values from 168 to 145) Paired T-Test Definition. The paired t-test gives a hypothesis examination of the difference between population means for a set of random samples whose variations are almost normally distributed. Subjects are often tested in a before-after situation or with subjects as alike as possible. The paired t-test is a test that the differences between the two observations are zero SPSS Statistics Output of the Dependent **T-Test** in SPSS Statistics. SPSS Statistics generates three tables in the Output Viewer under the title **T-Test**, but you only need to look at two tables: the **Paired** **Samples** Statistics table and the **Paired** **Samples** **Test** table. In addition, you will need to interpret the boxplots that you created to check for outliers and the output from the Shapiro-Wilk. Paired t-test data: group2 and group1 t = 4.4721, df = 9, p-value = 0.0007749 alternative hypothesis: true difference in means is greater than 0 95 percent confidence interval: 1.180207 Inf sample estimates: mean of the differences

Brief example of hand calculations for a paired-samples t-test

** This guide contains written and illustrated tutorials for the statistical software SAS**. Paired t tests are used to test if the means of two paired measurements, such as pretest/posttest scores, are significantly different. In SAS, PROC TTEST with a PAIRED statement can be used to conduct a paired samples t test Paired T-Test Statistic The paired t-test assumes that the paired differences, , are a simple random sample from a population of normally-distributed difference values that all have the same mean and variance. This assumption implies that the data are continuous, and their distribution is symmetric

- Like with the two independent-samples t-test, the paired-samples t-test follows the same steps for hypothesis testing: a. Define both H 0 and H 1 b. Set alpha (α, probability of a Type 1 Error) c. Identify decision rule (either for α, test statistic, or confidence interval) d. Calculate the test statistic (t ratio) e
- Standard error: Suppose that the assumptions of the paired sample $ t$ test hold: The difference scores are normally distributed in the population, with mean $\mu$ and standard deviation $\sigma$ The sample of difference scores is a simple random sample from the population of difference scores
- 2. Student's t-test One of the most popular approach for equality of population means is Student's t-test. This approach requires the observations in both samples are independent and normally distributed [3]. Let X 1 ˘( 1;˙2) and X 2 ˘( 2;˙2 2) be normal distributed random variables, then the t-test statistic is de ned as follows: (2.1.
- Paired samples t-test is another form of t-test which aims to test two means from those from the same sample group. The t-test is performed using the t-distribution as the basis for the development of the test. Paired t-test is performed to test 2 conditions using the mean test statistic of paired objects. Examples of frequently used uses: 1
- Reporting a paired sample t test 1. Reporting a Paired Sample t-test 2. Reporting a Paired Sample t-test Note - that the reporting format shown in this learning module is for APA. For other formats consult specific format guides. 3
- The Paired T Distribution, Paired T Test, Paired Comparison test, Paired Sample Test is a parametric procedure. Paired samples t-test are used when same group tested twice. It is often used in before and after designs where the same individuals are measured both before and after a treatment or improvement to see if changes occurred over time
- Paired t-test example. An instructor wants to use two exams in her classes next year. This year, she gives both exams to the students. She wants to know if the exams are equally difficult and wants to check this by looking at the differences between scores

the paired-samples test, there is the question of just how large ρ must be in order for the paired-samples t test to achieve more power than the independent-samples t test. Vonesh (1983) demonstrated that the paired-samples t test is more powerful than the independent-samples test when the correlation between the groups is .25 or larger. Paired T-Test vs Unpaired T-Test. The difference between the two statistical terms Paired T-test and Unpaired T-test is that in Paired T-Tests, you compare the differences between the paired measurements that have been deliberately matched whereas, in Unpaired T-Tests, you measure the difference between the means of two samples that do not have a natural pairing Hypothesis Tests: SingleSingle--Sample Sample tTests yHypothesis test in which we compare data from one sample to a population for which we know the mean but not the standard deviation. yDegrees of Freedom: The number of scores that are free to vary when estimating a population parameter from a sample df = N - 1 (for a Single-Sample t Test

Assumptions. This test assumes - The differences are of measurement variables.. Ordinal variables should not be analyzed using the paired t-test.. Sampling (or allocation) is random and pairs of observations are independent. Individual observations are clearly not independent - otherwise you would not be using the paired t-test - but the pairs of observations must be independen Note: If you do have all the data for your two related groups, as in our example above, but only the summarized data of the differences between your two related groups (i.e., the sample size, mean difference and standard deviation of the difference), Minitab can still run a paired t-test on your data. However, you will need to set up your data differently in order to do this Start studying Chapter 10: Paired Samples t Test. Learn vocabulary, terms, and more with flashcards, games, and other study tools

Standard Normal Distribution; Characteristics of the standard normal distribution; Area in tails of the distribution; Page 4. Transformation to Standard Normal; Page 5. Distribution of the Sample Mean; Using the t-table; Page 6. One-Sample Test of Means; One Sample t-test Using SAS: Page 7. Paired t-test; Paired t-test Using SAS: Reporting. Which section of a paired-samples t-test output can be ignored? a) The paired-samples correlations b) Descriptive statistics c) Means d) Standard deviations e) The paired differences f) The t, df and Sig. column

- When to use a paired sample t-test calculator? The paired samples t-test is called the dependent samples t test. It is used to compare the difference between two measurements where observations in one sample are dependent or paired with observations in the other sample. This is very typical in before and after measurements on the same subject
- A paired t-test can be run on a variable that was measured twice for each sample subject to test if the mean difference in measurements is significantly different from zero. For example, consider a sample of people who were given a pre-test measuring their knowledge of a topic. Then, they were given a video presentation about the topic, and were tested again afterwards with a post-test
- The T-Test For Paired Samples. More about the t-test for two dependent samples so you can understand in a better way the results delivered by the solver: A t-test for two paired samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)). More specifically, a t-test uses sample information to assess how plausible it is for difference \(\mu_1.
- sd is the standard deviation of of the paired differences. The sample size is the number of paired data samples. To illustrate this, let's now go over an example of a paired t-test scenario, which can be seen in real life. Let's say that we are running a new prep class
- Sample 1 has 20 df, Sample 2 has 22 df, so the total df is 42. We compare this t-score, 2.054 , to the critical value in the t- table, one-sided, 0.05 significance at 40 df (always round dow
- Target: the test compares the means of the same items in two different conditions or any others connection between the two samples when there is a one to one connection between the samples. The test uses the t distribution. more Two-tailed test example: Treatment is given to 50 people to reduce the cholesterol level. The expected reduction is.

- The repeated-measures t-test, also known as the paired samples t-test, is used to assess the change in a continuous outcome across time or within-subjects across two observations.There is only one group of participants with a repeated-measures t-test and their baseline or pretest mean and standard deviation serves as a control that is compared to their second or posttest mean and standard.
- Thus, in summary, an Unpaired 2-sample T-test takes as input 2 sample sets that are independent of each other, and the test's outputs follow a T-distribution. This is also abbreviated as an Unpaired T-test or Independent T-test. In contrast to the Unpaired 2-sample T-test, we also have the Paired 2-sample T-test
- The Independent-samples t-test. As suggested above, these tests are used to compare population means when separate samples were taken from the two populations. To keep this situation clearly distinct from paired-samples t-tests, I suggest that you write the null hypothesis (that the two population means are the same) as H 0: µ 1 = µ 2.
- This test is sometimes referred to as an independent samples t-test, or an unpaired samples t-test. Paired t-test A paired t-test is used to investigate the change in the mean of a population before and after some experimental intervention , based on a paired sample, i.e., when each subject has been measured twice: before and after treatment
- e whether to reject the null hypothesis, compare the t-value to the critical value. The critical value is t α/2, n-1 for a two-sided test and t α, n-1 for a one-sided test. For a two-sided test, if the absolute value of the t-value is greater than the critical value, you reject the null hypothesis
- This depends on what specific hypothesis you are trying to test. Note, however, that you can't mix multiple pairings for the same company. E.g. if for company XYZ you have data for one year pre- and post-merger as well as two-years pre- and post-merger, you can't use both pairs since this would violate the assumption of random, independent data in each sample (i.e. you can't have two.
- Use of the two sample or unpaired test is inappropriate. Stata's options for t-tests are one sample, two sample (with 2 options) and paired. When it comes to calculating the effect size, Stata provides two esize options. One is twosample, the other is unpaired. Note this is different from the t-test commands. With t-tests, we have paired and.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test. If you want to know only whether a difference exists, use a two-tailed test Perhaps one of the most widely used statistical hypothesis tests is the Student's t test. Because you may use this test yourself someday, it is important to have a deep understanding of how the test works. As a developer, this understanding is best achieved by implementing the hypothesis test yourself from scratch. In this tutorial, you will discover how to implement th Tutorial 2: Power and Sample Size for the Paired Sample t-test . Preface . Power is the probability that a study will reject the null hypothesis. The estimated probability is a function of sample size, variability, level of significance, and the difference between the null and alternative hypotheses. Similarly, the sample siz Python Paired Sample T-Test. In a Paired Sample T-Test, we will test whether the averages of 2 samples taken from the same population are different or not. Taking two sets of observations from the same population generates a pair of samples, reason why it is called the Paired Sample Test

- A paired samples t test will sometimes be performed in the context of a pretest-posttest experimental design. For this tutorial, we're going to use data from a hypothetical study looking at the effect of a new treatment for asthma by measuring the peak flow of a group of asthma patients before and after treatment
- Therefore, a paired-samples t-test conducted on two samples is equivalent to a one-sample t-test conducted on the difference scores (calculated on the two samples) compared against 0. Nota Bene: You will see that in the previous paired-samples t-test we obtained a t-statistic of -2.416, but here the t-statistic is positive
- Statistics: 1.2 Unpaired t-tests Rosie Shier. 2004. 1 Introduction An unpaired t-test is used to compare two population means. The following notation will be used throughout this leaﬂet: Group Sample size Sample mean Sample standard deviation 1 n 1 x¯ 1 s 1 2 n 2 x¯ 2 s 2 2 Procedure for carrying out an unpaired t-test
- Purpose of Paired Sample T-test Compare differences between two (2) dependent mean scores A paired-samples t-test is used when you have only one group of people (or companies, Or machines etc.) and we want collect data from them on two different occasions or under two different conditions
- Paired T-Test. Previously the observations in our two samples have been completely independent of one another. Perhaps we want to compare two related samples, e.g. a before and after test, we might use a paired T-test. This is calculated as follows: $$ t = \dfrac{\bar{d}}{s / \sqrt{n}} $

Similar to the z-t est, a t-test may apply to a single sample or two-sample situations. While the Standard Normal Distribution always has a mean of zero (0) and a standar d deviation of one (1. Paired samples t-tests typically consist of a sample of matched pairs of similar units, or one group of units that has been tested twice (a repeated measures t-test). A typical example of the repeated measures t -test would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure-lowering. 7: Paired Samples Data Paired samples vs. independent sample . This chapter considers the analysis of a quantitative outcome based on paired samples. Paired samples (also called dependent samples) are samples in which natural or matched couplings occur. This generates a data set in which each data point in one sample is uniquely paired to Paired t-tests can be conducted with the t.test function in the native stats package using the paired=TRUE option. Data can be in long format or short format. Examples of each are shown in this chapter. As a non-parametric alternative to paired t-tests, a permutation test can be used In all work with two-**sample** **t-test** the degrees of freedom or df is: The formula for the two **sample** **t-test** is: Males Females Males Females 70 87 165.9 212.1 71 89 210.3 203.5 72 90 166.8 210.3 76 94 182.3 228.4 77 97 182.1 206.2 78 99 218 203.2 80 101 170.1 224.9 102 202.6 All About Student's **t-test** Page 10 of 1

h = ttest(x) returns a test decision for the null hypothesis that the data in x comes from a normal distribution with mean equal to zero and unknown variance, using the one-sample t-test.The alternative hypothesis is that the population distribution does not have a mean equal to zero. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise Q: 2.An aptitude test consists of 10 questions with five alternatives each, of which one is correct. a.... A: It is given that there are 10 questions available and each question, having 5 alternative questions... Lecture Lecture+Activity Difference 82 88 -6 73 72 1 77 84 -7 71 74 -3 80 93 -13 SUM -28 N = 5 MEAN DIFFERENCE -5.6 MATCHED PAIRS OR DEPENDENT t- test In this chapter, you will learn the paired t-test formula, as well as, how to:. Compute the paired t-test in R.The pipe-friendly function t_test() [rstatix package] will be used.; Check the paired t-test assumptions; Calculate and report the paired t-test effect size using the Cohen's d.The d statistic redefines the difference in means as the number of standard deviations that separates.

t . test are the paired . t. test and the one-sample . t. test. There is a third form, the independent . t . test, and it will be discussed in Chapter 6, which concerns group research designs. We need to distinguish between the paired and one-sample . t. tests and between two general types of this test, one-tailed and two-tailed You can test for an average difference using the paired t-test when the variable is numerical (for example, income, cholesterol level, or miles per gallon) and the individuals in the statistical sample are either paired up in some way according to relevant variables such as age or perhaps weight, or the same people are used [

T-tests are hypothesis tests that assess the means of one or two groups. Hypothesis tests use sample data to infer properties of entire populations. To be able to use a t-test, you need to obtain a random sample from your target populations. Depending on the t-test and how you configure it, the test can determine whether The T-test for Two Independent Samples. More about the t-test for two means so you can better interpret the output presented above: A t-test for two means with unknown population variances and two independent samples is a hypothesis test that attempts to make a claim about the population means (\(\mu_1\) and \(\mu_2\)).. More specifically, a t-test uses sample information to assess how.

The **standard** deviation for the caffeine condition is 1.14 and for the no caffeine condition, also 1.14. The number of participants in each condition (N) is 5. **Paired** **Samples** **Test** Box . This is the next box you will look at. It contains info about the **paired** **samples** **t-test** that you conducted ** Thanks for your comment Teddy**. I do believe however that the t-test referred to as the t-test, by its construction, and as I wrote, assumes normality of the underlying observations in the population from which your sample is drawn (see the image I have now included in the bottom of the post, which is from Casella and Berger's book Statistical Inference) A paired-samples t test was conducted to evaluate the hypothesis that there was a difference in the mean ratings between the two brands of coffee. The mean rating for coffee A was 7 (SD = 1.155) and the mean rating for coffee B was 6.3 (SD = 1.059). The test was not significant, t(9) = 2.09, p > .05 Confidence Interval for paired t-test. In this tutorial we will discuss how to determine confidence interval for the difference in means for dependent samples

The paired samples t-test is used to compare the means between two related groups of samples. In this case, you have two values (i.e., pair of values) for the same samples. This article describes how to compute paired samples t-test using R software One-sample T-test Q(b): What is the probability that the new value (b) pooled standard error or variation within samples Two-sample T-test Q(x 1, x 2): Do samples A (x 1) and B (x 2) come from the For a paired t-test we only require that the pair-differences (A i-B Paired Sample t Test Example • We want to know if there is a difference in the salary for the same job in Boise, ID, and LA, CA. The salary of 6 employees in the 25th percentile in the two cities is given Example of hypotheses for paired and two-sample t tests. and conduct the two sample T test here, to see whether there's evidence that the sizes of be equal to 1.3 minus 1.6, 1.3 minus 1.6, all of that over the square root of, let's see, the standard deviation, the sample standard deviation from the sample from field A is 0.5.

The Paired-Sample t-Test (Chapter 9 in Zar, Fifth Edition) There is a type of experimental design that is referred to as a repeated measures design, because observations are made on an experimental unit prior to treatment, and again after treatment.The advantage of this is that it eliminates the variation and potential for bias that is present when assigning different experimental units for. ** to Zumbo and Jen nings (2002), it is still pos sible to use the paired samples t-test provided that: • The sample size is at l east 30 • The effect size (the mean of the di fference divided by**. A paired t-test just looks at the differences, so if the two sets of measurements are correlated with each other, the paired t-test will be more powerful than a two-sample t-test. For the horseshoe crabs, the P value for a two-sample t-test is 0.110, while the paired t-test gives a P value of 0.045 The paired samples t-test indicated that there was a statistically significant difference between stress scores on stress VAS 1 and 2 which suggests that the participants were significantly stressed t(48) = − 6.55, P < 0.01

Formula. The paired t-test statistics value can be calculated using the following formula: \[t = \frac{m}{s/\sqrt{n}} \] where, m is the mean differences; n is the sample size (i.e., size of d).; s is the standard deviation of d; We can compute the p-value corresponding to the absolute value of the t-test statistics (|t|) for the degrees of freedom (df): \(df = n - 1\) The Student's t-test is a statistical test that compares the mean and standard deviation of two samples to see if there is a significant difference between them.In an experiment, a t-test might be used to calculate whether or not differences seen between the control and each experimental group are a factor of the manipulated variable or simply the result of chance The paired t-test is the appropriate method when the researcher takes an experimental group, measures the baseline, subjects the members to an intervention, and then measures the results. Testing in a before-and-after manner like this (matched data or repeated measures) requires a different statistical technique than the typical student t-test » Paired t-Test. Paired t-Test in Excel When to Use the Paired t-Test. In the constant quest to reduce variation and improve products, companies need to evaluate different alternatives. A t-Test using two paired samples compares two dependent sets of test data. It helps determine if the means (i.e., averages) are different from each other

THE DEPENDENT-SAMPLES t TEST PAGE 2 EFFECT SIZE STATISTICS FOR THE DEPENDENT-SAMPLES t TEST Cohen's d (which can range in value from negative infinity to positive infinity) evaluates the degree (measured in standard deviation units) that the mean of the difference scores is equal to zero. If the calculated d equals 0, the mean of the difference scores is equal to zero h = ttest2(x,y) returns a test decision for the null hypothesis that the data in vectors x and y comes from independent random samples from normal distributions with equal means and equal but unknown variances, using the two-sample t-test.The alternative hypothesis is that the data in x and y comes from populations with unequal means. The result h is 1 if the test rejects the null hypothesis. Using SPSS for t Tests. This tutorial will show you how to use SPSS version 12.0 to perform one-sample t-tests, independent samples t-tests, and paired samples t-tests.. This tutorial assumes that you have: Downloaded the standard class data set (click on the link and save the data file) ; Started SPSS (click on Start | Programs | SPSS for Windows | SPSS 12.0 for Windows

2-Sample t-Test Overview A 2-sample t-test can be used to compare whether two independent groups differ. This test is derived under the assumptions that both populations are normally distributed and have equal variances. Although the assumption of normality is not critical (Pearson, 1931; Barlett, 1935 Since we are using sample standard deviations to estimate the population standard deviation, the test statistic from the t-distribution. The value of the test statistic is (84 - 75)/1.2583. This is approximately 7.15 Descriptive table showing the sample sizes n 1 =37 and n 2 =23, sample means x̄ 1 =67.86 and x̄ 2 =78.70, sample standard deviations s 1 =8.570 and s 2 = 7.600.. The below table is the Independent Sample Test Table, proving all the relevant test statistics and p-values Here's a simple check to determine if the paired t test can apply - if one sample can have a different number of data points from the other, then the paired t test cannot apply. Examples 'Student's' t Test is one of the most commonly used techniques for testing a hypothesis on the basis of a difference between sample means a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class Pair and rhs is 1 , a paired test is don