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Chapter 10: t test for two independent samples - Coggle Diagram
Chapter 10: t test for two independent samples
Take a story of testing goalkeepers in a soccer penalty shootout. A sample players shooting at a goalkeeper with red jerseys vs. a sample of players shooting at a goalkeeper with green jerseys. This is an example where two independent samples are compared using a t test.
Repeated measures research design or within subjects design involves the same group of participants. Ex. Before and after treatment design on same population.
Other examples include samples before and after treatment, political attitudes of one state vs another, and comparing two approaches to teaching math in schools.
Independent-measures or between subjects design uses two independent, separate samples and compares two groups of individuals. Examples include the goalkeeper jersey samples or comparing political attitudes of one state to another.
The null hypothesis of an independent measures test is µ₁ - µ₂ = 0. There is no difference in population means.
Independent t statistic formula: t=(M₁-M₂)-(µ₁ - µ₂)/s((M₁-M₂). The sample mean difference minus the population mean difference over the estimated standard error of the sample mean difference
Estimated standard error of M1-M2. It measures the average distance between the sample and population means. It also measures the average distance between two sample means when the null hypothesis is true.
The estimated error of M1-M2 is calculated by adding the variance of the two m
Pooled variance. Procedure to pool the variances of means of two populations to allow the larger population to have more weight in the estimated standard error of M1-M2.
Pooled variance formula is (SS1+SS2)/(df1+df2).
Estimated standard error is the square root of (Pooled variance/n1 + Pooled variance/n2)
Degrees of freedom is (n1-1)+(n2-1).
Steps to completing a t test statistic for two independent samples
Step 1. Find the pooled variance s²p of the two samples: (SS1+SS2)/(df1+df2)
Step 2: Use the pooled variance to calculate the estimated standard error, S(M1-M2)= √(s²p/n1+s²p/n2)
Step 3: Calculate the t statistic = ((M1-M2)-(µ1-µ2))/S(M1-M2).
Step 4: Make a decision. If the t statistic sits in the critical region, then we reject the null hypothesis.
Assumptions required for using an independent-measures t formula
Assumption 1: Observations from each sample must be independent
Assumption 2: The two populations from which samples are selected must be normal
Assumption 3: The two populations from which samples are selected must have equal variances.
Homogeneity of variance. Two populations being compared must have the same variance. When there is a large difference in sample sizes, we must assume equal variances or we cannot make a proper decision with respect to the null hyp
Hartley's F-max test. To check homogeneity of variance, take the largest variance of samples over the smallest variance of the samples. Look up with in a Table with alpha, df, and # of samples.
Effect size - Cohen's estimate d. (M1-M2)/√(SS1+SS2)/(df1+df2). Mean difference between samples over the square root of the pooled variance.
Cohen's d of 0.2 is small effect, 0.5 is medium, 0.8 is large.
r-squared is t-squared/(t-squared + df)
Confidence intervals are M1-M2 +/- t*s(M1-M2)