Identifying the Population and Variables By Dr.
In the t-test comparing the means of two independent samples, the following assumptions should be met: Each of the two populations being compared should follow a normal distribution.
This can be tested using a normality testsuch as the Shapiro—Wilk or Kolmogorov—Smirnov test, or it can be assessed graphically using a normal quantile plot. If using Student's original definition of the t-test, the two populations being compared should have the same variance testable using F-testLevene's testBartlett's testor the Brown—Forsythe test ; or assessable graphically using a Q—Q plot.
If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances. The data used to carry out the test should be sampled independently from the two populations being compared.
This is in general not testable from the data, but if the data are known to be dependently sampled that is, if they were sampled in clustersthen the classical t-tests discussed here may give misleading results.
Most two-sample t-tests are robust to all but large deviations from the assumptions. The simulated random numbers originate from a bivariate normal distribution with a variance of 1. Power of unpaired and paired two-sample t-tests as a function of the correlation. The simulated random numbers originate from a bivariate normal distribution with a variance of 1 and a deviation of the expected value of 0.
Two-sample t-tests for a difference in mean involve independent samples unpaired samples or paired samples. Paired t-tests are a form of blockingand have greater power than unpaired tests when the paired units are similar with respect to "noise factors" that are independent of membership in the two groups being compared.
Independent unpaired samples[ edit ] The independent samples t-test is used when two separate sets of independent and identically distributed samples are obtained, one from each of the two populations being compared. In this case, we have two independent samples and would use the unpaired form of the t-test.
Paired difference test Paired samples t-tests typically consist of a sample of matched pairs of similar unitsor 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 medication.
By comparing the same patient's numbers before and after treatment, we are effectively using each patient as their own control. That way the correct rejection of the null hypothesis here: Note however that an increase of statistical power comes at a price: This approach is sometimes used in observational studies to reduce or eliminate the effects of confounding factors.
Paired samples t-tests are often referred to as "dependent samples t-tests". Calculations[ edit ] Explicit expressions that can be used to carry out various t-tests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given.
Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed or two-tailed test. Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution.
If the calculated p-value is below the threshold chosen for statistical significance usually the 0.Student's t-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting .
Identify the population, variable of interest, and type of variable in the following: 1. The dean of the College of Arts and Sciences would like to determine the average weekly allowance of BS Applied Math Students.
Population Variable: _____ Type of Variable 2. Part 2 / Basic Tools of Research: Sampling, Measurement, Distributions, and Descriptive Statistics Chapter 9 Distributions: Population, Sample and Sampling Distributions I n the three preceding chapters we covered the three major steps in gathering and describing.
Creating Population Selection Rules Using A Variable Variables are defined in the Variable Rules Definition Form (GLRVRBL) and compiled in this application. You can click the Search icon to access the Variable Inqqyuiry Form (GLIVRBL) to search for a variable.
The population is the group of individuals you want to include in your study – the group you are proposing to study. They might be third grade students, teenagers in juvenile facilities, 10th grade students, boys and girls in preschool, Mothers, or maybe something bigger like studying differences in countries.
Variables: A population variable is a descriptive number or label associated with each member of a population. The values of a population variable are the various numbers (or labels) that occur as we consider all the members of the population. Values of variables that have been recorded for a population or a sample from a population constitute data.