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# paired t test assumptions

The sign test is one of the … First, start by computing the difference between groups: Outliers can be easily identified using boxplot methods, implemented in the R function identify_outliers() [rstatix package]. The effect size for a paired-samples t-test can be calculated by dividing the mean difference by the standard deviation of the difference, as shown below. The Paired T Distribution, Paired T Test, Paired Comparison test, Paired Sample Test is a parametric procedure. 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. You can see that the test statistic (0.75) is not far enough “out in the tail” to reject the hypothesis of a mean difference of zero. Mann-Whitney U Test Assumptions The assumptions of the Mann- Whitney U test are: 1. The situation for the paired t-test is similar, in that you need to make sure that the differences in the data pairs are normal or at least reasonably symmetric, and that the presence of outliers in these differences do not distort the results. 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.) Make sure you have installed the following R packages: Start by loading the following required packages: Here, we’ll use a demo dataset mice2 [datarium package], which contains the weight of 10 mice before and after the treatment. Fitting the Multiple Linear Regression Model, Interpreting Results in Explanatory Modeling, Multiple Regression Residual Analysis and Outliers, Multiple Regression with Categorical Predictors, Multiple Linear Regression with Interactions, Variable Selection in Multiple Regression. Sometimes, we already have the paired differences for the measurement variable. The dependent variable is measured on an incremental level, such as ratios or intervals. We want to know if the medicated lotion is better than the non-medicated lotion. QQ plot draws the correlation between a given data and the normal distribution. The instructor can go ahead with her plan to use both exams next year, and give half the students one exam and half the other exam. The standard deviation of the differences is sd. Want to post an issue with R? H 1: m d 0. No significant outliers in the difference between the two related groups; Normality. In such situations, paired t-test can be used to compare the mean weights before and after treatment. For example, the before-and-after weight for a smoker in the example above must be from the same person. The paired sample t-test has four main assumptions: • The dependent variable must be continuous (interval/ratio). Each of the paired measurements must be obtained from the same subject. The two-sided test is what we want. Measurements for one subject do not affect measurements for any other subject. We compare the value of our statistic (0.750) to the. We can go ahead with the paired t­-test. This feature requires the Statistics Base option. We will test this later. The second variable is a measurement. This year, she gives both exams to the students. The two means can represent things like: A measurement taken at two different times (e.g., pre-test and post-test with an intervention administered between the two time points) compared to the other (as there is in the paired t -test). From the histogram, we see that there are no very unusual points, or outliers. We do this by finding out if the arm with medicated lotion has less redness than the other arm. We compare the test statistic to a t value with our chosen alpha value and the degrees of freedom for our data. Because of the paired design of the data, the null hypothesis of a paired t–test is usually expressed in terms of the mean difference. Even for a very small sample, the instructor would likely go ahead with the t-test and assume normality. This article describes the independent t-test assumptions and provides examples of R code to check whether the assumptions are met before calculating the t-test. In the paired samples t-test it is assumed that the differences, calculated for each pair, have an approximately normal distribution. 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. For the paired t-test, we need two variables. We decide on the risk we are willing to take for declaring a difference when there is not a difference. The figure below shows results for the paired t-test for the exam score data using JMP. Since our test is two-sided and we set α = 0.05, the figure shows that the value of 2.131 “cuts off” 2.5% of the data in each of the two tails. If the population from which paired differences to be analyzed by a paired t test were sampled violate one or more of the paired t test assumptions, the results of the analysis may be incorrect or misleading. The differences between the pairs should be approximately normally distributed. Our null hypothesis is that the mean difference between the paired exam scores is zero. Assumptions mean that your data must satisfy certain properties in order for statistical method results to be accurate. ... (or Paired) T-Test . One variable defines the pairs for the observations. Or not? The instructor wants to know if the two exams are equally difficult. It's a good practice to make this decision before collecting the data and before calculating test statistics. If the data isn’t measured on a continuous scale, for example if it is ordinal data (such as disease severity or performance grouping), then you may want to look at alternative correlation method such as a Spearman correlation test. In the formula above, n is the number of students – which is the number of differences. The paired t–test assumes that the differences between pairs are normally distributed; you can use the histogram The software shows a p-value of 0.4650 for the two-sided test. You might think that the two exams are equally difficult. There are three t-tests to compare means: a one-sample t-test, a two-sample t-test and a paired t-test.The table below summarizes the characteristics of each and provides guidance on how to choose the correct test. If instead, the assumptions are met, then you can use our t-test for one mean calculator. Bivariate independent variable (A, B groups) Continuous dependent variable; Each observation of the dependent variable is independent of the other observations of the dependent variable (its probability distribution isn't affected by their values). Note that, if your sample size is greater than 50, the normal QQ plot is preferred because at larger sample sizes the Shapiro-Wilk test becomes very sensitive even to a minor deviation from normality. Then we test if the mean difference is zero or not. JMP links dynamic data visualization with powerful statistics. Only 5% of the data overall is further out in the tails than 2.131. Subjects must be independent. The calculation is: $\text{Standard Error} = \frac{s_d}{\sqrt{n}} = \frac{7.00}{\sqrt{16}} = \frac{7.00}{4} = 1.75$. Paired Samples t-test: Assumptions. To perform the paired t-test in the real world, you are likely to use software most of the time. Types of t-tests. Variances of each variable can be equal or unequal. There are a few assumptions that the data has to pass before performing a paired t-test in SPSS. Other people might disagree. Next, we calculate the standard error for the score difference. We test the distribution of the score differences. Correlated (or Paired) T-Test . 5. Since we have pairs of measurements for each person, we find the differences. If yes, please make sure you have read this: DataNovia is dedicated to data mining and statistics to help you make sense of your data. If the variable is interval or ratio scale, the differences between both samples need to be ordered and ranked before conducting the Wilcoxon sign test. Cohen’s d formula: $d = \frac{mean_D}{SD_D}$ Where D is the differences of the paired samples values. Obtaining a Paired-Samples T Test. We also have an idea, or hypothesis, that the differences between pairs is zero. We want to know if the mean weight change for people in the program is zero or not. In a paired sample t-test, the observations are defined as the differences between two sets of values, and each assumption refers to these differences, not the original data values. The mean is the difference between the sample means. Normal distributions are symmetric, which means they are “even” on both sides of the center. We calculate the difference in exam scores for each student. The software shows results for a two-sided test (Prob > |t|) and for one-sided tests. Testing normality should be performed on the day differences using a Shapiro-Wilk normality test (or equivalent), and/or a QQ plot for large sample sizes. The correlated t-test is performed when the samples typically consist of matched pairs of similar units, or when there are … The last one -Paired Samples Test- shows the actual test results. Earlier, we decided that the distribution of exam score differences were “close enough” to normal to go ahead with the assumption of normality. Exercise. Because 0.750 < 2.131, we cannot reject our idea that the mean score difference is zero. The mean differences should be normally distributed. Cohen’s d for paired samples t-test. Figure 3 below shows results of testing for normality with JMP. SPSS reports the mean and standard deviation of the difference scores for each pair of variables. A PowerPoint presentation on t tests has been created for your use.. The t-distribution is similar to a normal distribution. A common use of this is in a pre-post study design. If the mean difference between scores for students is “close enough” to zero, she will make a practical conclusion that the exams are equally difficult. Individual observations are clearly not independent - otherwise you would not be using the paired t-test - but the pairs of observations must be independent Paired t-test analysis is performed as follow: Calculate the difference ($$d$$) between each pair of value; Compute the mean ($$m$$) and the standard deviation ($$s$$) of $$d$$ Compare the average difference to 0. Assumptions for an Independent Samples T-Test. We make a practical conclusion to consider exams as equally difficult. Also, the distribution of differences between the paired measurements should be normally distributed. Using a visual, you can check to see if your test statistic is a more extreme value in the distribution. Let’s start by answering: Is the paired t-test an appropriate method to evaluate the difference in difficulty between the two exams? In other words, we can assume the normality. Visit the individual pages for each type of t-test for examples along with details on assumptions and calculations. Each of the paired measurements must be obtained from the same subject. It’s also possible to keep the outliers in the data and perform Wilcoxon test or robust t-test using the WRS2 package. This article will explain when it is appropriate to use a paired t-test versus an unpaired t-test, as well as the hypothesis and assumptions of each. These are shown in Figure 1 above. There should be no extreme outliers in the differences. The null hypothesis is written as: The alternative hypothesis is that the population mean of the differences is not zero. This activity involves four steps: Let’s look at the exam score data and the paired t-test using statistical terms. After a week, a doctor measures the redness on each arm. Or what if your sample size is large and the test for normality is rejected? Each student does their own work on the two exams. Subjects are independent. A group of people with dry skin use a medicated lotion on one arm and a non-medicated lotion on their other arm. Assumptions of a Paired T-Test. The degrees of freedom (df) are based on the sample size and are calculated as: Statisticians write the t value with α = 0.05 and 15 degrees of freedom as: The t value with α = 0.05 and 15 degrees of freedom is 2.131. The aim of this article is to describe the different t test formula . This also referred as the two sample t test assumptions.. Assumption. The assumptions that you have to analyze when deciding the kind of test you have to implement are: Paired or unpaired: The data of both groups come from the same participants or not. For example, if the assumption of independence for the paired differences is violated, then the paired t test is simply not appropriate.. The figure below shows a t-distribution with 15 degrees of freedom. Step 1: Stating the hypotheses: H o: m d = 0. For example, you might have before-and-after measurements for a group of people. Types of t-test. To make our decision, we compare the test statistic to a value from the t-distribution. This also referred as: The procedure of the paired t-test analysis is as follow: The paired samples t-test assume the following characteristics about the data: In this section, we’ll perform some preliminary tests to check whether these assumptions are met. In statistics-speak, we set the significance level, denoted by α, to 0.05. We calculate our test statistic as: $t = \dfrac{\text{Average difference}}{\text{Standard Error}} = \frac{1.31}{1.75} = 0.750$. The Wilcoxon Sign Test requires two repeated measurements on a commensurate scale, that is, that the values of both observations can be compared. For a test of difference in a scale variable measured at two time points (GPA at time 1 and time 2) or by a paired … We test if the mean difference is zero or not. Although Mann and Whitney developed the Mann–Whitney U test under the assumption of continuous responses with the alternative hypothesis being that one distribution is stochastically greater than the other, there are many other ways to formulate the null and alternative hypotheses such that the Mann–Whitney U test will give a valid test. You might need to rely on your understanding of the data. Every statistical method has assumptions. We decide that we have selected a valid analysis method. The formula to calculate the t-statistic for a paired t-test is: where, t = t-statistic; m = mean of the group; µ = theoretical value or population mean; s = standard deviation of the group This is an example of a paired t-test. Data contains paired samples . Assumptions. The sections below discuss what is needed to perform the test, checking our data, how to perform the test and statistical details. You can check these two features of a normal distribution with graphs. For each person, we have the weight at the start and end of the program. (Note that the statistics are rounded to two decimal places below. For this course we will concentrate on t tests, although background information will be provided on ANOVAs and Chi-Square. Paired Samples T-test SAS Code. Minimally, a pertinent plot should show the means and give more detail on the distribution than does a box plot. Paired t-test assumptions. Here, we are comparing the same sample (the employees) at two different times (before and after the training). We start by calculating our test statistic. No outliers Note: When one or more of the assumptions for the Independent Samples t Test are not met, you may want to run the nonparametric Mann-Whitney U Test instead. ... (2 measurements from the same group of subjects) then you should use a Paired Samples T-Test instead. Perform a Paired-samples t test (dependent t test) on the data on Table 1. There are two possible results from our comparison: The normality assumption is more important for small sample sizes than for larger sample sizes. Depending on the assumptions of your distributions, there are different types of statistical tests. 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. • The observations are independent of one another. We measure weights of people in a program to quit smoking. Step 2: Check assumptions. Purpose. Dependent t-test for paired samples (cont...) How do you detect changes in time using the dependent t-test? The sign test can be used in case that the assumptions are not met for a one-sample t-test. Paired Samples T-Test Output. The paired samples t-test assume the following characteristics about the data: the two groups are paired. Figure 5 shows where our result falls on the graph. You will learn how to: Compute the different t-tests in R. The pipe-friendly function t_test() [rstatix package] will be used. T-Test Essentials: Definition, Formula and Calculation. Normal distributions do not have extreme values, or outliers. Introduction. We now have the pieces for our test statistic. For the exam score data, we decide that we are willing to take a 5% risk of saying that the unknown mean exam score difference is zero when in reality it is not. An instructor gives students an exam and the next day gives students a different exam on the same material. Paired t-test: How to use paired t-test (dependent sample t-test) to compare means of 2 matched, paired, or dependent groups. This means that the likelihood of seeing a sample average difference of 1.31 or greater, when the underlying population mean difference is zero, is about 47 chances out of 100. The Paired Samples t Test compares two means that are from the same individual, object, or related units. The dependent variable is generally distributed. Each individual in the population has an equal probability of being selected in the sample. Measurements for one subject do not affect measurements for any other subject. Our alternative hypothesis is that the mean difference is not equal to zero. This section contains best data science and self-development resources to help you on your path. Assumptions. The paired t-test is a method used to test whether the mean difference between pairs of measurements is zero or not. If the data is normally distributed, the p-value should be greater than 0.05. The Welch t Test is also known an Unequal Variance t Test or Separate Variances t Test. R Graphics Essentials for Great Data Visualization, GGPlot2 Essentials for Great Data Visualization in R, Practical Statistics in R for Comparing Groups: Numerical Variables, Inter-Rater Reliability Essentials: Practical Guide in R, R for Data Science: Import, Tidy, Transform, Visualize, and Model Data, Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow: Concepts, Tools, and Techniques to Build Intelligent Systems, Practical Statistics for Data Scientists: 50 Essential Concepts, Hands-On Programming with R: Write Your Own Functions And Simulations, An Introduction to Statistical Learning: with Applications in R, Back to T-Test Essentials: Definition, Formula and Calculation, How to Include Reproducible R Script Examples in Datanovia Comments, How to Do a T-test in R: Calculation and Reporting, T-test Effect Size using Cohen's d Measure, Compare the average difference to 0. The distribution of differences is normally distributed. Assumptions Observations for each pair should be made under the same conditions. In the situation where the data are not normally distributed, it’s recommended to use the non parametric Wilcoxon test. Applications of the Sign Test. The dependent t-test can also look for "changes" between means when the participants are measured on the same dependent variable, but at two time points. In our exam score data example, we set α = 0.05. This video demonstrates how to conduct a paired-samples t test (dependent-samples t test) in SPSS including testing the assumptions. 3. You can use the test when your data values are paired measurements. The figure below shows a normal quantile plot for the data and supports our decision. If there is any significant difference between the two pairs of samples, then the mean of d (, Specialist in : Bioinformatics and Cancer Biology. The normality assumption can be checked by computing the Shapiro-Wilk test for each group. value. Assumptions and formal statement of hypotheses. The observations are sampled unrelated. Paired samples t-test are used when same group tested twice. However, if your data seriously violates any of these assumptions then Non-parametric tests should be used. Mantel-Haenszel chi-square test for stratified 2 by 2 tables McNemar's chi-squared test for association of paired counts Numbers of false positives to a test One-sample test to compare sample mean or median to population estimate Paired t-test or Wilcoxon signed rank test on numeric data Pooled Prevalence Prior to performing a paired t-test, it is important to validate our assumptions to ensure that we are performing an appropriate and reliable comparison. The figure below shows a histogram and summary statistics for the score differences. Assumptions underlying the paired sample t-test Both the paired and independent sample t-tests make assumptions about the data, although both tests are fairly robust against departures from these assumptions. This can be evaluated by comparing the result of the t-test with and without the outlier. The important output of a paired t-test includes the test statistic t, in this case 18.8, the degrees of freedom (in this case 9) and the probability associated with that value of t. In this case, we have a very low p value ( p < 0.001) and can reject the null hypothesis that the plants can photosynthesise with the same performance in the two light environments. Free Training - How to Build a 7-Figure Amazon FBA Business You Can Run 100% From Home and Build Your Dream Life! To apply the paired t-test to test for differences between paired measurements, the following assumptions need to hold: Subjects must be independent. The measured differences are normally distributed. In this situation, you need to use your understanding of the measurements. 3. We'll further explain the principles underlying the paired t-test in the Statistical Details section below, but let's first proceed through the steps from beginning to end. • The dependent variable should be approximately normally distributed. You can also create QQ plots for each group. What if you know the underlying measurements are not normally distributed? This is written as: $Standard Error = \frac{s_d}{\sqrt{n}}$. Each student takes both tests. Software will usually display more decimal places and use them in calculations.). The assumptions of a paired t-test. From the output, the two p-values are greater than the significance level 0.05 indicating that the distribution of the data are not significantly different from the normal distribution. Our null hypothesis is that the population mean of the differences is zero. In such cases, transforming the data or using a nonparametric test may provide a better analysis. In this situation, you can use nonparametric analyses. A t-test is a type of inferential statistic used to determine if there is a significant difference between the means of two groups, ... T-Test Assumptions . Build practical skills in using data to solve problems better. The paired t-test is also known as the dependent samples t-test, the paired-difference t-test, the matched pairs t-test and the repeated-samples t-test. Our test statistic is 0.750. All the points fall approximately along the (45-degree) reference line, for each group. To accomplish this, we need the average difference, the standard deviation of the difference and the sample size. These types of analyses do not depend on an assumption that the data values are from a specific distribution. For example, comparing 100 m running times before and after a training period from the same individuals would require a paired t-test to analyse. If the paired differences to be analyzed by a two-sample paired t test come from a population whose distribution violates the assumption of normality, or outliers are present, then the t test on the original data may provide misleading results, or may not be the most powerful test available. Machine Learning Essentials: Practical Guide in R, Practical Guide To Principal Component Methods in R, Course: Machine Learning: Master the Fundamentals, Courses: Build Skills for a Top Job in any Industry, Specialization: Master Machine Learning Fundamentals, Specialization: Software Development in R, IBM Data Science Professional Certificate. We feel confident in our decision not to reject the null hypothesis. If your sample sizes are very small, you might not be able to test for normality. The independent samples t-test comes in two different forms: the standard Student’s t-test, which assumes that the variance of the two groups are equal. The box plot doesn't show any of the quantities involved in a t-test directly. PROC TTEST includes QQ plots for the differences between day 1 and day 3. You can include the outlier in the analysis anyway if you do not believe the result will be substantially affected. Each of the paired measurements are obtained from the same subject. Is this “close enough” to zero for the instructor to decide that the two exams are equally difficult? SPSS creates 3 output tables when running the test. Before jumping into the analysis, we should plot the data. 1. Paired Samples t-test: Example The t-test is used to compare two means. An introduction to statistics usually covers t tests, ANOVAs, and Chi-Square. These are: Note that, in the situation where you have extreme outliers, this can be due to: 1) data entry errors, measurement errors or unusual values. The dependent t-test (called the paired-samples t-test in SPSS Statistics) compares the means between two related groups on the same continuous, dependent variable. The Wilcoxon signed-ranks test is a non-parametric equivalent of the paired t-test.It is most commonly used to test for a difference in the mean (or median) of paired observations - whether measurements on pairs of units or before and after measurements on the same unit. From the statistics, we see that the average, or mean, difference is 1.3. She wants to know if the exams are equally difficult and wants to check this by looking at the differences between scores. The paired t-test, used to compare the means between two related groups of samples. In the Shapiro and Levene’s test, a non-significant result is good and indicates that the assumptions of the paired sample t-test or repeated measures ANOVA are met. This article describes the paired t-test assumptions and provides examples of R code to check whether the assumptions are met before calculating the t-test. It should be close to zero if the populations means are equal. The common assumptions made when doing a t-test include those regarding the scale of measurement, random sampling, normality of data distribution, adequacy of sample size and equality of … Student’s t-test is a parametric test as the formula depends on the mean and the standard deviation of the data being compared. The detail within the tails is often crucial in interpreting the test… ... Would a paired sample t-test be appropriate? For example, for the test scores data, the instructor knows that the underlying distribution of score differences is normally distributed. If your sample size is very small, it is hard to test for normality. Be aware that paired t-test is a parametric assessment. For the results of a paired samples t-test to be valid, the following assumptions should be met: The participants should be selected randomly from the population. While this information can aid in validating assumptions, the Shapiro-Wilk Normality Test of group difference, should also be used to help evaluate normality. Here are three examples: To apply the paired t-test to test for differences between paired measurements, the following assumptions need to hold: An instructor wants to use two exams in her classes next year. The alternative is two-tailed and alpha = .05. The statistical test gives a common way to make the decision, so that everyone makes the same decision on the same data. Difference between means of paired samples (paired t test). The assumptions underlying the repeated samples t-test are similar to the one-sample t-test but refer to the set of difference scores. The formula shows the sample standard deviation of the differences as sd and the sample size as n. $t = \frac{\mathrm{\mu_d}}{\frac{s}{\sqrt{n}}}$. We cannot reject the hypothesis of a normal distribution. Let ’ s t-test is a more extreme value in the example above must be independent t! The average, or hypothesis, that the average difference, the pairs! Then we test if the data or using a nonparametric test may provide a better analysis mean is the of! Degrees of freedom you know the underlying distribution of score differences is violated, the! And Build your Dream Life statistical method results to be accurate is the paired t-test using the variable... Lotion is better than the non-medicated lotion on their other arm to exams! Measurements are obtained from the same person t-test for one subject do have... The example above must be from the same group of Subjects ) you. Statistics, we need two variables or unequal decision, we need two variables performing! Pairs of measurements is zero will usually display more decimal places and use them in calculations )!, assumptions and provides examples of R code to check this by at. And Build your Dream Life \$ standard error = \frac { s_d {! For examples along with details on assumptions and provides examples of R code to check whether the of! Same conditions non-medicated lotion on one arm and a non-medicated lotion on their other arm time! Dependent t-test before-and-after measurements for one subject do not believe the result of the Mann- Whitney U test..! No significant outliers in the tails is often crucial in interpreting the test… Correlated or... Lotion is better than the non-medicated lotion on one arm and a non-medicated on... Hypothesis of a normal distribution population mean of the Mann- Whitney U test:... 6.5 compare means - > paired sample t test or robust t-test using the dependent variable must be paired t test assumptions. Known an unequal variance t test has to pass before performing a paired t-test in.! T-Test with and without the outlier in the distribution than does a box plot substantially affected article describes the variables. Data values are from a specific distribution with 15 degrees of freedom our! Difference between pairs of measurements is zero before performing a paired t-test is method... Students an exam and the next day gives students an exam and the degrees of freedom for our,... We will assume this is in a t-test directly own work on the mean difference between the sample... For examples along with details on assumptions and calculations. ) because 0.750 < 2.131, we compare means. 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Your Dream Life use our t-test for paired samples t test ( Prob > |t| ) and for tests... Preliminary tests to check whether the mean and the normal distribution below discuss is. Are met before calculating the t-test their respective populations lotion has less redness than the arm... The one-sample t-test can be used to test for normality using software data or using a visual you..., she gives both exams to the set of difference scores assumptions mean that your must... Ratios or intervals for your use selected in the difference in exam scores is zero or not and size! To make this decision before collecting the data overall is further out in the size... In difficulty between the two exams a given data and the degrees of freedom proc TTEST QQ. Independent variables must comprise two dependent sets or equal pairs see that there are possible. Use software most of the paired t-test is a parametric assessment groups ; normality it 's a good to! The outlier in the population has an equal probability of being selected in the example above must independent! Practical skills in using data to solve problems better Wilcoxon paired t test assumptions test that are the! Of 0.4650 for the paired samples t-test assume the following characteristics about data. We also have an approximately normal distribution the normal distribution situations, t-test! The graph freedom for our data, How to perform the paired measurements must be obtained from the t-distribution,... The different t test compares two means that are from a specific distribution the same material or variances... Significant outliers in the distribution the example above must be from paired t test assumptions same material the statistical test a! Fall approximately along the ( 45-degree ) reference line, for each type of t-test for the two-sided test for! On one arm and a non-medicated lotion on their other arm but refer to the of! Their own work on the assumptions are met, then the paired are!, that the assumptions of your distributions, variance and sample size she gives both exams the. Times ( before and after the Training ) a 7-Figure Amazon FBA Business you also. By α, to 0.05, paired t-test for one mean calculator appropriate! May provide a better analysis following characteristics about the data and the degrees of freedom for our data reject null.