Empirical Direction in Design and Analysis

By Norman H. Anderson | Go to book overview

NOTES
0.1.2a
A third answer to the proposal to select representative samples of subjects is that they are practically impossible to achieve. Individual behavior is highly unpredictable in novel situations.
0.1.3a
Although MSerror= s2is an unbiased estimate of σ 2, its square root, s, is not an unbiased estimate of σ. This bias is not ordinarily pertinent; in particular, it does not affect the t test or confidence intervals.
0.1.3b
The confidence interval will of course differ from sample to sample. However, 95% of the confidence intervals around the sample mean will contain the population mean. Hence you are entitled to 95% confidence that the confidence interval calculated from your one particular sample contains the population mean (see further The Concept of Confidence, page 94).
0.1.3c
The confidence interval also adds information about the size of the effect—measured jointly by the width of the confidence interval and the distance between the end of the interval and 0.
0.4.1a
Formula for Normal Distribution. Although the mathematical formula for the normal distribution is not needed to understand anything in this book, it is given here to show that it really does exist. For a normal distribution with mean μ and standard deviation, σ, this formula is

This formula gives the probability, Y, of each value of X, as illustrated in Figure 0.1 where X is women's heights. In this formula, it denotes the ratio of the circumference of a circle to its diameter, and e denotes the base for natural logarithms.

This formula shows that all normal distributions have the same overall shape; they differ only in their mean, μ, and standard deviation, σ. The multiplier, 1/σ √2π, makes the total area under the normal curve equal to 1, as it must to be a probability distribution.

For the curious, this formula contains three famous numbers. Two are π and e, workhorse numbers that pop up everywhere in mathematics. Both are transcendental numbers, so called because they are not the roots of any polynomial equation with rational coefficients. √2 is an irrational number—not expressible as the ratio of two whole numbers—and hence not really a number to the ancient Greeks. The discovery that √2 was irrational caused a crisis in the Pythagorean religion analogous to, although lesser than, the crisis in the Catholic church caused by Galileo's discoveries.

0.2.1a
Probability theory also includes the NOT rule: Prob(NOT A) = 1 Prob(A).
0.3a
Runs of consecutive events present some interesting aspects of probability thinking. A couple has four girls in a row and are expecting their fifth child. Will it be another girl? Since height and other physical characteristics are correlated across siblings, it seems reasonable to expect the same for sex. On this plausible argument, the probability that the fifth child will also be a girl is greater than ½

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