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Week 4, The estimated standard error is used as an estimate of the actual…
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The estimated standard error is used as an estimate of the actual standard error when the value of s is unknown. It is computed from the sample variance or sample standard deviation and provides an estimate of the standard distance between a sample mean M and the population mean μ.
The t statistic is used to test hypotheses about an unknown population mean, μ, when the value of s 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.
A t distribution is the complete set of t values computed for every possible random sample for a specific sample size (n) or a specific degrees of freedom (df). The t distribution approximates the shape of a normal distribution.
A confidence interval is an interval, or range of values centered around a sample statistic. The logic behind a confidence interval is that a sample statistic, such as a sample mean, should be relatively near to the corresponding population parameter. Therefore, we can confidently estimate that the value of the parameter should be located in the interval near to the statistic.
Total variability minus variability not explained by the treatment effect is equal to the amount of variability accounted for by the treatment. The difference between these two values, points, is the amount of variability that is accounted for by the treatment effect. This value is usually reported as a proportion or percentage of the total variability.
many researchers prefer to identify the calculated value as an “estimated d” or name the value after one of the statisticians who first substituted sample statistics into Cohen’s formula (e.g., Glass’s g or Hedges’s g).
Correlation is a statistical technique that is used to measure and describe the relationship between two variables.
This measure of effect size is related to correlation, r, and we now have an opportunity to demonstrate the relationship. Specifically, we compare the independent-measures t test (Chapter 10) and a special version of the Pearson correlation known as the point-biserial correlation .
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When both variables (X and Y) measured for each individual are dichotomous, the correlation between the two variables is called the phi-coefficient .
In a positive correlation , the two variables tend to change in the same direction: As the value of the X variable increases from one individual to another, the Y variable also tends to increase; when the X variable decreases, the Y variable also decreases.
In the scatter plot, the values for the X variable are listed on the horizontal axis and the Y values are listed on the vertical axis. Each individual is represented by a single point in the graph so that the horizontal position corresponds to the individual’s X value and the vertical position corresponds to the Y value. The value of a scatter plot is that it allows you to see any patterns or trends that exist in the data.
This line, called an envelope because it encloses the data, often helps you to see the overall trend in the data.
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The squared correlation measures the proportion of variability in the data that is explained by the relationship between X and Y. It is sometimes called the coefficient of determination
In a negative correlation , the two variables tend to go in opposite directions. As the X variable increases, the Y variable decreases. That is, it is an inverse relationship.
The Pearson correlation measures the degree and the direction of the linear relationship between two variables.
The calculation of the Pearson correlation requires one new concept: the sum of products of deviations, or SP. This new value is similar to SS (the sum of squared deviations), which is used to measure variability for a single variable. Now, we use SP to measure the amount of co-variability between two variables. The value for SP can be calculated with either a definitional formula or a computational formula.
When the Pearson correlation formula is used with data from an ordinal scale (ranks), the result is called the Spearman correlation .
The statistical technique for finding the best-fitting straight line for a set of data is called regression
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to determine the total error between the line and the data, we add the squared errors for all of the data points. The result is a measure of overall squared error between the line and the data
The equation that determines the linear equation that provides the best prediction of Y values is called the regression equation for Y.
Occasionally, however, researchers standardize the scores by transforming the X and Y values into z-scores before finding the regression equation. The resulting equation is often called the standardized form of the regression equation and is greatly simplified compared to the raw-score version. The simplification comes from the fact that z-scores have standardized characteristics.
The standard error of estimate gives a measure of the standard distance between the predicted Y values on the regression line and the actual Y values in the data.
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In the general linear equation, the value of b is called the slope . The slope determines how much the Y variable changes when X is increased by one point.
The value of a in the general equation is called the Y-intercept because it determines the value of Y when X = 0.