Ch. 13: Comparing Two Populations: Independent Samples

Confidence Interval Formula

Upper limit= M1-M2 + (s{M1-M2} x t{α/2}) Lower limit= M1-M2 - (s{M1-M2} x t{α/2})

Standard Error of the Sampling Distribution of M1-M2

(but just 1 value for σ )

T Statistic for Independent Samples

-compute pooled variance -compute estimated standard error of the sampling distribution -compute t statistic

Increasing Power of the Independent Sample t Test

-increase alpha -increase effect size -decrease variability (increase n) -appropriate use of directional alternative hypothesis

Estimate of the Standard Error, s(M1-M2)

-used to compute the t statistic for independent samples

Hypothesis Testing Using T

1. Make sure assumptions are met 2. Hypotheses 3. Make the sampling distribution of the test statistic (z= T-μT/σT), which will make a normal distribution 4. Set alpha and the decision rule 5. Sample and compute the test statistic (compute μT, σT, then z statistic) 6. Decide and draw conclusions

Calculating Power of Independent Sample t-Test

1. Propose an effect size based on a value for μ1-μ2. Either compute it or pick a standard value (.2, .5, .8) Given my alpha, sample size, and alternative hypothesis, what is the probability that I will reject the null if my proposal for the effect size is correct? 2. Compute 𝛿 3. Look in table for the column with your type of alternative and alpha, and the row with your 𝛿

Sample Size Analysis for Independent Sample t Test

1. Specify desired power 2. Give an estimate of the population effect size (d) using standard values or the formula Given my alpha and type of alternative hypothesis, what sample size do I need to obtain the desired power if my estimate of the effect size is correct? 3. Find in the table the 𝛿 corresponding to the alpha and alternative and power you're looking for 4. Compute n

Assumptions of Rank-Sum T Test

1. The two samples are independent 2. The two samples are obtained using independent (within-sample) random sampling 3. Both samples have 8 or more observations (so the sampling distribution of the T statistic is approximately normally distributed). If only smaller samples are available, you should use the Mann-Whitney U statistic 4. The data are ordinal, interval, or ratio

Hypothesis Testing with the Sampling Distribution of the Independent Sample t Statistic

Goal: determine whether the means of 2 populations are the same or different 1. Make sure assumptions are met 2. Hypotheses 3. Sampling Distributions (of t statistic based on the hypotheses) 4. Set alpha and decision rule (reject H0 if p-value is < α/2, or if t> or < critical t value) 5. Sample and compute the test statistic (compute sample means, pooled variances, estimate of the standard error, then t statistic) 6. Decide and draw conclusions

The Rank-Sum Statistic, T

How to compute the T Statistic: 1. Combine the n1+n2 scores from the two samples and order them from smallest to largest 2. Assign the rank of 1 to the smallest score and the rank of n1+n2 to the largest score. If there are ties, assign the average rank to all of the tied scores. 3. Sum the ranks corresponding to the scores in the first sample. This sum is the statistic T. 4. Check your work: the sum of the ranks in both groups should equal (n1+n2)(n1+n2+1)/2 When the null hypothesis is correct (the two populations have the same distributions), then both the high and low ranks should be equally distributed across the two samples. If the null is incorrect, the one of the samples should have higher scores and the other should have lower, so the T is one sample will be higher than the other. To determine whether or not the null is correct, we determine if the T statistic is very large or very small by comparing it to the sampling distribution of the statistic.

Logic of Statistical Comparison of Two Populations

In order to guess a population mean (to be able to do hypothesis testing), it can help to make inferences about two population means and ask if the two population means differ. -in experiments, ask if the control and experimental group means differ -use "independent sample t-test" to analyze the difference between 2 population means

The Rank-Sum Test for Independent Samples

Nonparametric procedure that corresponds to the independent-sample t test; used to compare 2 populations when 2 independent samples from them are available. When assumptions for independent sample t test are grossly violated, you have to resort to this nonparametric test. Doesn't require any assumptions about the population distributions, so the test is less powerful and the null is less specific (so you don't know exactly how the samples differ if it's rejected)

Hypotheses of Independent Sample t Test

Null Hypothesis: must propose a specific value for μ1-μ2 (the mean of the sampling distribution of M1-M2). The symbol for the difference is Δ0, and it's typically 0. H0: μ1-μ2= Δ0= 0 Alternative Hypothesis: proposes difference between population means is greater than, less than, or not equal to 0 (Δ0) -usually choose to use the nondirectional

Sampling Distribution of the Difference Between Sample Means (Independent Samples)

Null hypothesis: μ1-μ2=0 (no difference between two population means) Test statistic: t statistic based on M1-M2 Sampling distribution of M1-M2 characteristics: 1. The mean of the sampling distribution of M1-M2= μ1-μ2. 2. When the two populations have the same variance, then the standard deviation of the sampling distribution of M1-M2 (standard error) is given by the formula, and the standard error decreases as sample size increases 3. If the two populations are normally distributed, then the sampling distribution of M1-M2 will be normally distributed. As sample size increases, the sampling distribution approaches a normal distribution

Hypotheses of Rank-Sum T Test

Null: H0: the two populations have identical relative frequency distributions Nondirectional Alternative: H1: the two populations do not have identical relative frequency distributions Directional Alternative: H1: the scores in Population 1 tend to exceed the scores in Population 2

Estimating the Difference Between Two Population Means

The statistic M1-M2 is an unbiased point estimator for μ1-μ2, but it's probably not exact, so interval estimation is the way to go. 1. Make sure assumptions are met (relatively normal distribution, same population variance or sample size, the samples are independent of one another, independent random sampling was used, interval or ratio data) 2. Set the Confidence Level: 1-α (the probability that the interval will contain the real μ1-μ2) 3. Obtain the random samples 4. Construct the confidence interval -calculate M1-M2 -calculate sp² so you can calculate standard error (s{M1-M2}) -find t{α/2} (t statistic with n1 + n2-2df that has α/2 of the distribution above it) 5. Interpretation: x% confident that the interval includes the real value of μ1-μ2. Any null hypothesis outside the range would be rejected.

Sampling Distribution of the T Statistic

When the two sample sizes are both more than 7 and the null is correct, -the mean of the sampling distribution μT= n1(n1+n2+1)/2 -The standard deviation of the sampling distribution σT= √(n1)(n2)(n1+n2+1)/12 -the sampling distribution of T is approximately normally distributed -can convert the T statistic into a z score and use the table z= (T-μT)/σT When the null is correct, then the T from the sample will be approximately equal to μT, and the z score will be close to 0 When the null is incorrect, then T will be much larger or smaller than μT, and z will be much larger or smaller than 0

Naturally Occurring Population

a population that is present without any intervention by the investigator -ex) amount of alcohol consumed in small towns

The t Distribution for Independent Samples

convert the sampling distribution of M1-M2 into the distribution of the t statistic so we can use tabled values Characteristics: 1. t distribution with n1+n2-2 degrees of freedom (b/c each sample has n-1 dfs) 2. symmetric about the mean of 0

Assumptions of Independent-Sample t Test

if these assumptions can't be met, use the rank-sum test 1. The two populations are normally distributed (or relatively so, since the t statistic is robust) 2. The two populations have the same variance (unless the sample sizes are almost equal, then it doesn't matter) 3. The scores in the 2 samples are independent of one another 4. Each of the samples must be obtained using independent (within-sample) random sampling from its population 5. The data is measured using an interval or ratio scale

N formula

n= 2(𝛿/d)²

Effect Size (d) Estimation

requires guesses for σ (common population variance) and a specific difference for the difference between the population means (Δs) d= |Δs-Δ0|/σ

Hypothetical Population

the observations don't exist until they are actually measured -ex) achievement scores of students who use study method 1 vs. study method 2 -can take random samples even though the populations don't really exist, only the samples exist. Comparing the random samples will tell you about the respective populations. -cheap, effective, and more ethical than attempting to create populations -treated the same statistically as naturally occurring populations

Pooled Variance

the weighted average of s1² and s2² -give more weight to the larger sample by multiplying the variances by n-1 (degrees of freedom in the sample variance) -estimate of the common population variance (the two populations are assumed to have the same variance) -used to estimate the standard error of the sampling distribution

Dependent Samples

two or more random samples are dependent when the scores included in one random sample are systematically related to the scores included in another random sample -often when the same or related individuals contribute scores to the 2 samples -aka: paired samples, correlated samples, matched samples, repeated measures, within-subject sampling -can use more powerful statistical tests with these -samples should be obtained by independent (within-sample) random sampling

Independent Samples

two or more random samples are independent when the scores included in one random sample are unrelated to the scores included in the other random sample -different and unrelated people contribute the scores in the 2 samples -this chapter is only about statistical techniques you can apply to independent samples! -different from independent (within-sample) random sampling (measurement of any observation in the sample shouldn't affect measurement of any other observation in the sample); but should be obtained by this type of sampling

𝛿 Formula

𝛿= d√(n1)(n2) / (n1+n2)