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T-Statistic, Correlation, . - Coggle Diagram
T-Statistic
used to test hypotheses about an unknown population mean, μ, when the value of o is unknown. The formula for the t statistic has the same structure as the z-score formula, except that the t statistic uses the estimated standard error in the denominator.
Estimate the standard error using the sample data. The estimated standard error can then be used to compute a new statistic that is similar to the z-score. The new statistic is called a t statistic and it can be used to conduct a new kind of hypothesis test; used in situations for which the population standard deviation is unknown; When the variance (or standard deviation) for the population is not known, we use the corresponding sample value in its place.
Estimated Standard Error: sM; used as an estimate of the real standard error (oM) when the value of (o) is unknown; computed from the sample variance or sample standard deviation; provides an estimate of the standard distance between a sample mean M and the population mean μ.
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Degrees of Freedom: df=n-1; describe the number of scores in a sample that are independent and free to vary.
The greater the value of df for a sample, the better the sample variance,s2, represents the population variance, o2, and the better the t statistic approximates the z-score
The t Distribution: you can compute the t statistic for every sample and the entire set of t values will form a t distribution; The larger the value of df is, the more closely the t distribution approximates a normal distribution.
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Influence of Sample Size and Sample Variance: The two factors that determine the size of the standard error are the sample variance, , and the sample size, n.
Measuring Effect Size for the t Statistic: Estimated Cohen’s d = mean difference / sample standard deviation
Measuring the Percentage of Variance Explained,
r2: An alternative method for measuring effect size is to determine how much of the variability in the scores is explained by the treatment effect.
To measure the size of the treatment effect we calculate deviations from the mean and the sum of squared deviations, SS, two different ways.
SS including treatment effect-SS without treatment effect = # of points of variability between each SS
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percentage of variance accounted for by the treatment (r2)
A measure of effect size that determines what portion of the variability in the scores can be accounted for by the treatment effect.
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Correlation
A correlation measures the relationship between two variables, X and Y and are described by:
1. Direction: either positive or negative. A positive relationship means that X and Y vary in the same direction. A negative relationship means that X and Y vary in opposite directions. The sign of the correlation (+ or −) specifies the direction.
2. Form: most common form for a relationship is a straight line. However, special correlations exist for measuring other forms. The form is specified by the type of correlation used; the Pearson correlation measures linear form.
3. Strength or consistency: The numerical value of the correlation measures the strength or consistency of the relationship. A correlation of 1.00 indicates a perfectly consistent relationship and 0.00 indicates no relationship at all. For the Pearson correlation, (or −1.00) means that the data points fit perfectly on a straight line.
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To evaluate the strength of a relationship, you square the value of the correlation.
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measures the portion of the variability in one variable that can be determined using the relationship with the second variable.
A partial correlation measures the linear relationship between two variables by eliminating the influence of a third variable by holding it constant.
The Spearman correlation (rS): measures the consistency of direction in the relationship between X and Y; the degree to which the relationship is one-directional, or monotonic
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Point-Biserial Correlation: used to measure the strength of the relationship when one of the two variables is dichotomous.
dichotomous variable is coded using values of 0 and 1, regular Pearson formula is applied.
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