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Chapter 10 The t Test for Two Independent Samples, Chapter 12 An Overview…
Chapter 10 The t Test for Two Independent Samples
Single-sample techniques are used only occasionally in real search as most research studies require the comparison of two or more sets of sample data. Single-sample techniques use one sample as the basis for drawing conclusions about one population.
Independent measures research design (also known as between-subjects design) uses completely separate groups (two samples).
Hypotheses for an independent-measures test: The goal of this kind of study is to evaluate the mean difference between two populations. Researchers must use subscripts to differentiate two populations. the difference between the means is u1-u2.
The null hypothesis is Ho: u1-u2=0 (meaning that there is no difference in the population means. This could also be stated as u1=u2
The alternative hypothesis states that there is a mean difference between the two populations. H1:u1-u2 does not equal 0. (meaning, there is a difference)
The Formulas for an independent-measures hypothesis test use another version of the t statistic. The formula for this new t statistic has the same general structure as the earlier introduced t statistic formula.
The original t-statistic is called the
single-sample t statistic
and the new formula is the
independent-measures t statistic
.
The basic structure is the same for both the independent-measures and the single-sample hypothesis tests. t= (actual difference between sample data and the hypothesis/expected difference between sample data and hypothesis with no treatment effect). However the difference is that the Independent-measures t is basically a two-sample t that doubles all elements of the single-sample t formulas.
This form of t formula adds ((M1-M2) - (u1-u2)) / S (M1-M2) to the overall formula.
The estimated standard error the standard error in the denominator measures how much error is expected between these sample statistic and the population parameter. In the single sample t formula, the standard error measures the amount of error expected for a sample mean and is represented by the symbol sm.
For the independent measures t formula, the standard error measures the amount of error that is expected a between a sample mean difference (M1-M2) and the population mean difference (u1-u2). The standard error for the sample mean difference is represented by the symbol s(m1-m2).
Two ways to Interpreting the estimated standard error: Number 1- It measures the standard distance between (M1-M2) and (u1-u2). Number 2- When the null hypothesis is true, it measures the stand, or average size of (M1-M2). That is, it measures how much difference is reasonable to expect between the two sample means.
When Calculating the Estimated Standard Error consider the two following: 1) Each of the two sample means represents its own population mean, but in each case there is some error. 2) For the independent-measures t statistic, we want to know the total amount of error involved in using two sample means to approximate two population means. To do this, we will find the error from each sample separately and then add the two errorstogether.
Estimated Standard Error: using the pooled variance in place of the individual sample variances, we can obtain an unaided measure of the standard error for a sample mean difference.
Repeated-measures design (also known as within-subjects design) uses two sets of data that are obtained from the same group of participants.
Pooled Sample: combining the two sample variances into a single value. Why would you need to do this? Because the two sample sizes are different and thus the formula is biased.
In equal sample sizes the pooled difference is exactly halfway between the two sample variance, and is simply the average of the two sample variances.
In unequal sample sizes the pooled variance is not located halfway between the two sample variances. Instead, the pooled value is closer to the variance for the larger sample. The larger sample carries more weight, and when computing the weight you'll use the degrees of freedom. The larger sample has a larger degrees of freedom (df) so it carries more weight.
Hypothesis Tests with Independent-Measures t Statistics
Step 1: State the hypothesis and select the alpha level.
Step 2: Locate the critical region.
Step 3: Obtain the data and compute the test statistic.
Step 4: Make a decision.
Directional Hypotheses and One-tailed tests
Step 1: State the hypotheses and select the alpha level.
Step 2: Locate the critical regiona
Step 3: Collect the data and calculate the test statistic.
Step 4: Make a Decision
Assumptions underlying the independent measures t-formula
1: The observations within each sample must be independent.
2: The two populations from which the samples are selected must be normal.
3: The two populations from which the samples are selected must have equal variances.
Chapter 12 An Overview of Analysis of Variance
Analysis of variant (ANOVA) is a hypothesis-testing procedure that is used to evaluate mean differences between two or more treatments (or populations).
It's an inferential procedure and uses sample data as the basis for drawing general conclusions about populations.
ANOVA has an advantage over t tests because it can be used to compare two or more treatments and thus provides researcher with much greater flexibility in designing experience and interpreting results.
Typical ways in which ANOVA would be used? The goal of the analysis is to determine whether the mean differences observed among the sample provide enough evidence to conclude that there are mean differences amount 3 samples or populations. This is determined by:
1) There really are no differences between the populations. The observed differences between the sample means are caused by random, unsystematic factors (sampling error) that differentiate one sample from another.
Aka the null hypothesis
2) The populations really do have different means, and these population mean differences are responsible for causing systematic differences between the sample means.
aka the alternative hypothesis
The variable (independent or quasi-independent) that designates the groups being compared is called a
factor
.
The individual conditions or values that make up a factor are called the
levels
of the factor.
Testwise alpha leve
l is the risk of a Type I error, or alpha level, for an individual hypothesis test.
When an experiment involves several different hypothesis tests, the
experiment wise alpha
level is the total probability of a Type I error that is accumulated from all of the individual tests in the experiment. Typically, the experiment wise alpha level is substantially greater than the value of alpha used for any one of the individual tests.
F-ratio
F = (variance between sample means) / (variance expected with no treatment effect)
F-ratio has the same basic structure as the t statistic but is based on variance instead of sample mean difference.
The variance in the numerator of the F-ratio provides a single number that measures the differences amount all of the sample means.
The variance in the denominator of the F-ratio measures the mean differences that would be expected if there is no treatment effect.
Thus, the t statistic and the F-ratio provide the same basic information. In each case a large value for the test statistic provides evidence that the sample mean differences are larger than would be expected if there was no treatment effects.
Between Treatments Variance simply measures how much difference exists between the treatment conditions.
1) the differences between treatments are not caused by any treatment effect but are simply the naturally occurring, random differences that exist between one sample and another.
2) The differences between treatments have been caused by the treatment effects. For example, if treatments really do affect performance, then scores in one treatment should be systematically different from scores in another condition.
Within Treatments variance provides a measure of how big the differences are when Ho is true.
The F-ratio: The Test Statistic for ANOVA compares treatments and within treatments.
Formula for independent-measures ANOVA, the f-ratio has the following structure: F(variance between treatments/variance within treatments) = (differences including any treatments/differences with no treatments)
When we express each component of variability in terms of its sources, the structure of the F-ration is: F= (systematic treatment effects = random and unsystematic differences) / (random and unsystematic differences)
The value obtained for the F-ration helps determine whether any treatment effects exist. Consider the following two possibilities:
1) when there are no systematic treatment effects, the differences between treatments are entirely caused by random, unsystematic factors.
2) When the treatment does have an effect, causing systematic differences between samples, then the combination of systematic and random differences in the numerator should be larger than the random differences alone in the denominator.