bus 310 chapter 10: two sample tests and one way anova

confidence interval estimate for the difference between two proportions

(p1 − p2) ± Zα/2√[p1(1 − p1)]/n1 + [p2(1 − p2)]/n2

cases in which related data is used

1) when one takes repeated measurements from the same set of items or individuals 2) when one matches items or individuals according to some characteristic in these situations, one examines the difference between the two related values rather than the individual values themselves whether matched samples or repeated measurements is used, the objective is to study the difference between two measurements by reducing the effect of the variability that is due to the items or individuals themselves

one way anova f test assumption

1. randomness and independence of the samples selected - most critical assumption; validity of any experiment depends on random sampling or a randomization process; to avoid biases in outcomes, one needs to select random samples from the c groups or use a randomization process to randomly assign the items to the c levels of the factor - selecting a random sample or randomly assigning the levels ensures that a value from one group is independent of any other value in the experiment - departures from this assumption can seriously affect inferences made using the ANOVA results 2. normality of c groups from which the samples are selected - the one way anova test is fairly robust against departures from the normal distribution; as long as the distributions are not extremely different from a normal distribution, the level of significance of the anova f test is usually not greatly affected, particularly for large samples - one can assess normality of each of the c samples by constructing a normal probability plot or a boxplot 3. homogeneity of variance (the variances of the c groups are equal) - if each group has the same sample size, inferences based on the f distribution are not seriously affected by unequal variances - whenever possible, groups should have equal sample sizes because with unequal sample sizes, unequal variances can have a serious effect on inferences made using anova results when only the normality assumption is violated, one can use the kruskal-wallis rank test, a nonparametric procedure when homogeneity of variance assumption is violated, one can use procedures similar to those used in separate variance test when both the normality and homogeneity of variance assumptions ahve been violated, one needs to use an appropriate data transformation that both normalizes the data and reduces differences in variances or use amore general nonparametric procedure

null and alternative hypothesis for pooled variance t test

H0: μ1=μ2 or μ1−μ2 = 0 H1: μ1≠μ2 or μ1−μ2 ≠ 0

null and alternative for test of equality for population variances

H0: σ1^2 = σ2^2 H1: σ1^2 ≠ σ2^2 reject the null if the f stat is greater than the upper tail critical value from the f distribution with n1-1 d.o.f. in the numerator and n2-1 d.o.f. in the denom

null and alternative hypotheses of test for difference in means of two related populations

H0:μD = 0 H1:μD≠0

mean squares among (MSA)

MSA = SSA/(c − 1)

mean square total (MST)

MST = SST / (n-1)

mean square within (MSW)

MSW = SSW/(n − c)

variance of two samples when n1=n2

Sp^2 = (S1^2+S2^2)/2

variance for difference between two means

Sp^2 = [(n1 − 1) S1^2 + (n2 − 1) S2^2] / [(n1 − 1) + (n2 − 1)]

z stat for difference of proportions

ZSTAT = [(p1 − p2) − (π1 − π2)] / √[¯p(1 − ¯p)(1/n1 + 1/n2)]

analysis of variance (ANOVA)

allows statistical comparison among samples taken from many populations - comparison is typically the result of an experiment while they analyze variation, the purpose of ANOVA is to reach conclusions about possible differences among the means of each group, analogous to the hypothesis tests; each ANOVA design uses samples that represent each group and subdivides the total variation observed across all samples (all groups) toward the goal of analyzing possible differences among the means of each group subdivision (partitioning) works is a function of the design being used but total variation represented by the quantity sum of squares total (SST) will always be the starting point

levels

analogous to the categories of a categorical variable actual locations

matched samples

another type of related data between populations; items or individuals are paired together according to some characteristic of interest ex: in test marketing a product in two different advertising campaigns, sample of test markets can be matched on the basis of the test market population size and/or demographic variables - by accounting for differences in test market population size and or demographic variables, one can better measure the effects of the two different advertising campaigns

null of ANOVA

assuming that the c groups represent populations whose values are randomly and independently selected, follow a normal distribution and have equal variances, the null of no differences in the population means H0:μ1 = μ2 = ⋯ = μc alternative that not all the c population means are equal: H1:Not all μj are equal (where j = 1,2,...,c).

pooled variance t test

assumption that the variances in the two populations are equal (otherwise use a separate variance test) if one assumes that the random samples are independently selected from two populations and that the populations are normally distributed and have equal variances, pooled variance t test can be used to determine whether there is a significant difference between the means - if populations do not differ greatly from a normal distribution, on can still use the pooled variance t test especially if sample sizes are large enough (typically greater or equal to 30 for each sample) called pooled variance because test statistic pools or combines, the two sample variances S1^2 and S2^2 to calculate Sp^2, the best estimate of the variance common to both populations under the assumption that the two population variances are equal

d.o.f. anova

because one compares c groups, there are c-1 degrees of freedom associated with the sum of squares among groups; because each of the c groups contributes nj-1 d.o.f., there are n-c degrees of freedom associated with the sum of squares within groups - in addition, there are n-1 degrees of freedom associated with the sum of squares total because you are comparing each value Xij to the grand mean based on all n values

sum of squares among groups (SSA)

calculates the among group variation by summing the squared differences between the sample mean of each group and the grand mean weighted by sample size nJ in each group SSA = c∑j = nj(¯Xj −¯X)^2

one way ANOVA

completely randomized design analyses a single factor two part process: 1) determine if there is a significant difference among group means; if one rejects the null that there is no difference among the means, one proceeds with a second method that seeks to identify the groups whose means are significantly different from the other group means to analyze variation towards the goal of determining possible differences among the group means, you partition the total variation into variation that is due to differences among the groups and variation that is due to the differences within groups - symbol n represents the number of values in all groups and the symbol c represents the number of groups

determining assumption of two equal variances

complicated because when sampling from two independent populations, one almost always does not know the standard deviation of either population using sample variances, one can test whether the two population variances are equal

critical values of the f distribution

depends on the degrees of freedom in the two samples - the numerator d.o.f. are the d.o.f. for the first sample and the denominator d.o.f. are the d.o.f. of the second sample

f test for the ratio of two variances

determines whether there is evidence of a difference in the two population variances; the results of that test can help one decide which of of the t tests, pooled variance or separate variance is more appropriate

total variation (SST)

df = n-1

mean squares

dividing each of these sums of squares by its respective degrees of freedom computes three variances - another term for variance that is used in the analysis of variance - because the mean square is equal to the sum of squares divided by the degrees of freedom, a mean square can never be negative

level of significance for pooled variance t test

for a given level of significance in a two tail test, one rejects the null if the t stat test statistic is GREATER than the upper tail critical value from the distribution or if the t stat test stat is LESS than the lower tail critical value from the t distribution

decision rule for difference in means of two related populations

for a two tail test with a given level of significance, you reject the null if the t stat is greater than the upper tail critical value or if the t stat is less than the lower critical value from the t distribution

separate variance t test

for situations in which two independent populations can be assumed to be normally distributed but cannot be assumed to have equal variances sometimes results from pooled variance and separate variance t tests conflict because the assumption of equal variances is violated - f test for the ratio of two variances

two sample tests

hypothesis testing that compares statistics from samples selected from two populations

paired t test for the mean difference

if one can assume that the difference scores are randomly and independently selected from a population that is normally distributed, one can use the paired t test for the mean difference in related populations to determine whether there is significant population mean difference - follows the t distribution with n-1 degrees of freedom - although the paired t test is normally distributed, one can use this test as long as the sample size is not very small and the population is not highly skewed because this test is robust tSTAT = ¯D − μD / SD/√n

anova summary table

includes entries for the sources of variation (among groups, within groups and total), the d.o.f., the sum of squares, the mean squares (the variances and the computed f stat test stat; may also include the p value (probability of having an f stat value as large as r larger than the one computed, given the null is true - p value enables one to reach conclusions about the null without needing to refer to the table of critical values of the f distribution; if less than the level of significance, you reject the null

confidence interval estimate for the difference between two means

instead of, or in addition to, testing for the difference between the means of two independent populations to develop a confidence interval estimate of the difference in the means (¯X1 − ¯X2)± tα/2 √Sp^2 (1/n1 + 1/n2) one can be 95% confident that the difference in means is between the two values; because interval does not include zero, one rejects the null hypothesis of no difference between the means of the two populations

groups

levels provide the basis of comparison by dividing the variable under study into groups

grand mean

mean of all the values in the group combined

among group variation (SSA)

measures differences from group to group df = c-1

within group variation (SSW)

measures random variation df = n-c

f test normality assumption

one assumes that each of the two populations is normally distributed; the f test is very sensitive to the normality assumption - if boxplots or normal probability plots suggest even mild departure from normality for either of the two populations, one should not use the f test and instead use the levene test or a nonparametric approach in testing for the equality of variances as part of assessing the appropriateness of the pooled variance t test procedure, the F test is a two tail test with a/2 in the upper tail; however when one examines variability in situations other than the pooled variance t test, the f test is often one tail

null hypothesis for z test for the difference between two proportions

states that the two population proportions are equal (π1 = π2) because the pooled estimate for the population proportion is based on the null hypothesis, you combine or pool the two sample proportions to compute ¯p, an overall estimate of the common population proportion the estimate is equal to the number of items of interest in the two samples (X1 + X2) divided by the total sample size from the two samples (n1+n2) H0: π1 = π2 or π1 − π2 = 0 H1: π1≠π2 or π1 − π2≠0

anova test of equality of population means

subdivide the total variation in the values into two parts- that which is due to variation among the groups and that which is due to variation within the groups - total variation is represented by the sum of squares total (SST) because the population means of the c groups are assumed to be equal under the null hypothesis, one calculates the total variation among all the values by summing the squared differences between each individual value and the grand mean

pooled variance t test t stat

t stat = (¯X1 − ¯X2) − (μ1 − μ2) / √S2p (1/n1 + 1/n2) t distribution of: n1+n2 -2 degrees of freedom

z test for the difference between proportions

test stat based on the difference between two sample proportions (p1-p2) and the test stat approx follows a standardized normal distribution for large enough sample sizes ZSTAT = [(p1 − p2) − (π1 − π2)] / √[¯p(1 − ¯p)* (1/n1 + 1/n2)] ¯p = [X1 + X2] / [n1 + n2] p1 = X1/n1 p2 = X2/n2

levene test

tests the equality of variances to test the homogeneity of variance null: H0: o1^2 = o2^2 = oc^2 alternative: H1: not all o^2 are equal (j = 1,2,3, ..., c) to test the null of equal variances, one first calculates the absolute value of the difference between each value and the median of the group; the one performs anova using these absolute differences typically using a significance of 0.05

factor

the basis for an ANOVA experiment "how much of a factor is in store location in determining mobile electronics sales?"

tukey kramer procedure

the one way anova f test indicates if there is a difference among the c groups - when a difference is discovered, the next step is to construct multiple comparisons to test the null tukey-kramer multiple comparisons procedure for one way anova: determines which of the c means are significantly different - procedure enables one to simultaneously make comparisons between all pairs of groups

f distribution

the test for difference between the variances of two independent populations is based on the ratio of the two sample variances; if one assumes that each population is normally distributed, then the sampling distribution of the ratio S1^2/S2^2 is distributed as the f distribution unlike normal and t distributions which are symmetric, the f distribution is right skewed for the f test for the ratio of two variances, the sample with the larger sample variance is defined as the first sample and the sample with the smaller sample variance is defined as the second sample - the population from which the first sample is drawn is population 1 and the population for the second sample is population 2 f stat = S1^2/S2^2 follows a distribution of n1-1 and n2-1 degrees of freedom

sum of squares within groups (SSW)

the within group variation measures the difference between each value and the mean of its own group and sums the squares of these differences over all groups SSW = c∑j = nj∑i = (Xij − ¯Xj)^2

f test for differences among more than two means

to determine if there is a significant difference amount the c group means, use the f test for differences among more than two means - the f distribution is right skewed with a minimum value of 0; if the null is true and there are no differences among the c group means, MSA, MSW and MST will provide estimates of the overall variance in the population f stat = MSA/MSW follows an f distribution with c-1 numerator d.o.f. and n-c d.o.f. null: H0: μ1 = μ2 = ⋯ = μc alternative: H1: Not all μj are equal (where j = 1,2,...,c)] for a given level of significance, reject the null if the fstat is greater than the upper tail critical value from the f distribution with c-1 numerator d.o.f. and n-c denominator d.o.f. if the null is true, the f stat is expected to be approx equal to 1 because both the numerator and denominator mean square terms are estimating the overall variance in the population; if null is false (and there are differences in the group means), the f stat is expected to be larger than 1 because the numerator - MSA is estimating the differences among groups in addition to the overall variability in values while the denominator MSW is measuring only the overall variability in the values

using the f test to determine whether two variances are equal

to determine whether to use a pooled variance t test or the separate variance t test, one first tests the equality of the two population variances null and alternative: H0: σ1^2 = σ2^2 H1: σ1^2 ≠ σ2^2 0.05 level of significance, the rejection region in the upper tail contains 0.025 of the distribution

difference scores

to test for the mean difference between two related populations, one treats the difference scores, each Di as values from a single sample

two tail and one tail z tests for the difference between population proportions

two tail test: H0: π1 = π2 H1: π1 ≠ π2 one tail test H0: π1 ≥ π2 H1: π1 < π2 one tail test H0: π1 ≤ π2 H1: π1 > π2 for a given level of significance, one rejects the null if the z stat is greater than the upper critical value from the standardized normal distribution or if the z stat test stat is less than the lower tail critical value from standardized normal distribution

repeated measurements

when one takes repeated measurements on the same items or individuals, one assumes that the same items or individuals will behave alike if treated alike - objective is to show that any differences between two measurements of the same items or individuals are due to different treatments that have been applied to the items or individuals using repeated measurements enables one to answer questions like "do prices for the same items differ between two retailers?"; by collecting prices of the same items from both sellers, one creates two related samples and can use a test that is more than powerful - those tests use two independent samples that most likely will not contain the same sample of times - that means that differences observed might be due to one sample having products that are inherently costlier than the other

normality assumption

when the two populations have equal variances, the pooled variance t test is robust (not sensitive) to moderate departures from the assumption of normality, provided that the sample sizes are large - in such situations, one can use the pooled variance t test without serious effect on its power, the probability that one correctly rejects a false null - for cases in which one cannot assume that both populations are normally distributed, two alternatives exist: use a NON PARAMETRIC measure like the Wilcoxon rank sum test that does not depend on the assumption of normality for two populations or use a NORMALIZING TRANSFORMATION on each of the values before using the pooled variance t test

confidence interval estimate for the mean difference

¯D ± tα/2*SD/√n