For which of the following cases is a ttest of independent means not suitable?
The paired sample ttest, sometimes called the dependent sample ttest, is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. In a paired sample ttest, each subject or entity is measured twice, resulting in pairs of observations. Common applications of the paired sample ttest include casecontrol studies or repeatedmeasures designs. Suppose you are interested in evaluating the effectiveness of a company training program. One approach you might consider would be to measure the performance of a sample of employees before and after completing the program, and analyze the differences using a paired sample ttest. Discover How We Assist to Edit Your Dissertation ChaptersAligning theoretical framework, gathering articles, synthesizing gaps, articulating a clear methodology and data plan, and writing about the theoretical and practical implications of your research are part of our comprehensive dissertation editing services.
HypothesesLike many statistical procedures, the paired sample ttest has two competing hypotheses, the null hypothesis and the alternative hypothesis. The null hypothesis assumes that the true mean difference between the paired samples is zero. Under this model, all observable differences are explained by random variation. Conversely, the alternative hypothesis assumes that the true mean difference between the paired samples is not equal to zero. The alternative hypothesis can take one of several forms depending on the expected outcome. If the direction of the difference does not matter, a twotailed hypothesis is used. Otherwise, an uppertailed or lowertailed hypothesis can be used to increase the power of the test. The null hypothesis remains the same for each type of alternative hypothesis. The paired sample ttest hypotheses are formally defined below:
The mathematical representations of the null and alternative hypotheses are defined below:
Note. It is important to remember that hypotheses are never about data, they are about the processes which produce the data. In the formulas above, the value of \(\mu_d\) is unknown. The goal of hypothesis testing is to determine the hypothesis (null or alternative) with which the data are more consistent. AssumptionsAs a parametric procedure (a procedure which estimates unknown parameters), the paired sample ttest makes several assumptions. Although ttests are quite robust, it is good practice to evaluate the degree of deviation from these assumptions in order to assess the quality of the results. In a paired sample ttest, 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 paired sample ttest has four main assumptions:
Level of MeasurementThe paired sample ttest requires the sample data to be numeric and continuous, as it is based on the normal distribution. Continuous data can take on any value within a range (income, height, weight, etc.). The opposite of continuous data is discrete data, which can only take on a few values (Low, Medium, High, etc.). Occasionally, discrete data can be used to approximate a continuous scale, such as with Likerttype scales. IndependenceIndependence of observations is usually not testable, but can be reasonably assumed if the data collection process was random without replacement. In our example, it is reasonable to assume that the participating employees are independent of one another. NormalityTo test the assumption of normality, a variety of methods are available, but the simplest is to inspect the data visually using a tool like a histogram (Figure 1). Realworld data are almost never perfectly normal, so this assumption can be considered reasonably met if the shape looks approximately symmetric and bellshaped. The data in the example figure below is approximately normally distributed. Histogram of an approximately normally distributed variable.OutliersOutliers are rare values that appear far away from the majority of the data. Outliers can bias the results and potentially lead to incorrect conclusions if not handled properly. One method for dealing with outliers is to simply remove them. However, removing data points can introduce other types of bias into the results, and potentially result in losing critical information. If outliers seem to have a lot of influence on the results, a nonparametric test such as the Wilcoxon Signed Rank Test may be appropriate to use instead. Outliers can be identified visually using a boxplot (Figure 2). Boxplots of a variable without outliers (left) and with an outlier (right).ProcedureThe procedure for a paired sample ttest can be summed up in four steps. The symbols to be used are defined below:
The four steps are listed below:
determine whether the results provide sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis. InterpretationThere are two types of significance to consider when interpreting the results of a paired sample ttest, statistical significance and practical significance. Statistical SignificanceStatistical significance is determined by looking at the pvalue. The pvalue gives the probability of observing the test results under the null hypothesis. The lower the pvalue, the lower the probability of obtaining a result like the one that was observed if the null hypothesis was true. Thus, a low pvalue indicates decreased support for the null hypothesis. However, the possibility that the null hypothesis is true and that we simply obtained a very rare result can never be ruled out completely. The cutoff value for determining statistical significance is ultimately decided on by the researcher, but usually a value of .05 or less is chosen. This corresponds to a 5% (or less) chance of obtaining a result like the one that was observed if the null hypothesis was true. Practical SignificancePractical significance depends on the subject matter. It is not uncommon, especially with large sample sizes, to observe a result that is statistically significant but not practically significant. In most cases, both types of significance are required in order to draw meaningful conclusions. Statistics Solutions can assist with your quantitative analysis by assisting you to develop your methodology and results chapters. The services that we offer include: Data Analysis Plan Edit your research questions and null/alternative hypotheses Write your data analysis plan; specify specific statistics to address the research questions, the assumptions of the statistics, and justify why they are the appropriate statistics; provide references Justify your sample size/power analysis, provide references Explain your data analysis plan to you so you are comfortable and confident Two hours of additional support with your statistician Quantitative Results Section (Descriptive Statistics, Bivariate and Multivariate Analyses, Structural Equation Modeling, Path analysis, HLM, Cluster Analysis) Clean and code dataset Conduct descriptive statistics (i.e., mean, standard deviation, frequency and percent, as appropriate) Conduct analyses to examine each of your research questions Writeup results Provide APA 6th edition tables and figures Explain chapter 4 findings Ongoing support for entire results chapter statistics Please call 7274424290 to request a quote based on the specifics of your research, schedule using the calendar on this page, or email [email protected] When would using an independent means tUse an independent samples t test when you want to compare the means of precisely two groups—no more and no less! Typically, you perform this test to determine whether two population means are different.
In which of the following cases we can apply tThe Paired Samples t Test is commonly used to test the following: Statistical difference between two time points. Statistical difference between two conditions. Statistical difference between two measurements.
What is tThe ttest for independent means compares the difference between two independent sample means to an expectation about the difference in the population. For this test, we do not need to know the population parameters.
Which of the following is required in the use of independent tThe independent ttest requires that the dependent variable is approximately normally distributed within each group.
