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From Wikipedia
In statistics, a contingency table (also referred to as cross tabulationor cross tab) is often used to record and analyze the relation between two or morecategorical variables. It displays the (multivariate) frequency distribution of the variables in a matrix format.
The term contingency table was first used by Karl Pearson in "On the Theory of Contingency and Its Relation to Association and Normal Correlation", part of the Drapers' Company Research Memoirs Biometric Series I published in 1904.
Example
Suppose that we have two variables, sex (male or female) and handedness (right or lefthanded). Further suppose that 100 individuals are randomly sampled from a very large population as part of a study of sex differences in handedness. A contingency table can be created to display the numbers of individuals who are male and righthanded, male and lefthanded, female and righthanded, and female and lefthanded. Such a contingency table is shown below.
The numbers of the males, females, and right and lefthanded individuals are called marginal totals. The grand total, i.e., the total number of individuals represented in the contingency table, is the number in the bottom right corner.
The table allows us to see at a glance that the proportion of men who are righthanded is about the same as the proportion of women who are righthanded although the proportions are not identical. The significance of the difference between the two proportions can be assessed with a variety of statistical tests including Pearson's chisquare test, the Gtest, Fisher's exact test, and Barnard's test, provided the entries in the table represent individuals randomly sampled from the population about which we want to draw a conclusion. If the proportions of individuals in the different columns vary significantly between rows (or vice versa), we say that there is a contingency between the two variables. In other words, the two variables are not independent. If there is no contingency, we say that the two variables are independent.
The example above is the simplest kind of contingency table, a table in which each variable has only two levels; this is called a 2 x 2 contingency table. In principle, any number of rows and columns may be used. There may also be more than two variables, but higher order contingency tables are difficult to represent on paper. The relation between ordinal variables, or between ordinal and categorical variables, may also be represented in contingency tables, although such a practice is rare.
Measures of association
The degree of association between the two variables can be assessed by a number of coefficients: the simplest is the phi coefficient defined by
 \phi=\sqrt{\frac{\chi^2}{N}},
where Ï‡^{2} is derived from Pearson's chisquare test, and N is the grand total of observations. Ï† varies from 0 (corresponding to no association between the variables) to 1 or 1 (complete association or complete inverse association). This coefficient can only be calculated for frequency data represented in 2 x 2 tables. Ï† can reach a minimum value 1.00 and a maximum value of 1.00 only when every marginal proportion is equal to .50 (and two diagonal cells are empty). Otherwise, the phi coefficient cannot reach those minimal and maximal values.
Alternatives include the tetrachoric correlation coefficient (also only applicable to 2 × 2 tables), the contingency coefficientC, and CramÃ©r'sV.
C suffers from the disadvantage that it does not reach a maximum of 1 or the minimum of 1; the highest it can reach in a 2 x 2 table is .707; the maximum it can reach in a 4 x 4 table is 0.870. It can reach values closer to 1 in contingency tables with more categories. It should, therefore, not be used to compare associations among tables with different numbers of categories. Moreover, it does not apply to asymmetrical tables (those where the numbers of row and columns are not equal).
The formulae for the C and V coefficients are:
 C=\sqrt{\frac{\chi^2}{N+\chi^2}} and
 V=\sqrt{\frac{\chi^2}{N(k1)}},
k being the number of rows or the number of columns, whichever is less.
C can be adjusted so it reaches a maximum of 1 when there is complete association in a table of any number of rows and columns by dividing C by \sqrt{\frac{k1}{k}} (recall that C only applies to tables in which the number of rows is equal to the number of columns and therefore equal to k).
The tetrachoric correlation coefficient assumes that the variable underlying each dichotomous measure is normally distributed. The tetrachoric correlation coefficient provides "a convenient measure of [the Pearson productmoment] correlation when graduated measurements have been reduced to two categories." The tetrachoric correlation should not be confused with the Pearson productmoment correlation coefficient computed by assigning, say, values 0 and 1 to represent the two levels of each variable (which is mathematically equivalent to the phi coefficient). An extension of the tetrachoric correlation to tables involving variables with more than two levels is the polychoric correlation coefficient.
The Lambda coefficientis a measure the strength of association of the cross tabulations when the variables are measured at thenominal level. Values range from 0 (no association) to 1 (the theoretical maximum possible association). Asymmetric lambdameasures the percentage improvement in predicting the dependent variable.
In mathematics, the inverse trigonometric functions or cyclometric functions are the inverse functions of the trigonometric functions, though they do not meet the official definition for inverse functions as their ranges are subsets of the domains of the original functions. Since none of the six trigonometric functions are onetoone (by failing the horizontal line test), they must be restricted in order to have inverse functions.
For example, just as the square root function y = \sqrt{x} is defined such that y^{2} = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such that sin(y) = x; for example, sin(0) = 0, but also sin(Ï€) = 0, sin(2Ï€) = 0, etc. It follows that the arcsine function is multivalued: arcsin(0) = 0, but also arcsin(0) = Ï€, arcsin(0) = 2Ï€, etc. When only one value is desired, the function may be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x) will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometric functions.
The principal inverses are listed in the following table.
If x is allowed to be a complex number, then the range of y applies only to its real part.
The notations sin^{−1}, cos^{−1}, etc. are often used for arcsin, arccos, etc., but this convention logically conflicts with the common semantics for expressions like sin^{2}(x), which refer to numeric power rather than function composition, and therefore may result in confusion between multiplicative inverse and compositional inverse.
In computer programming languages the functions arcsin, arccos, arctan, are usually called asin, acos, atan. Many programming languages also provide the twoargument atan2 function, which computes the arctangent of y / x given y and x, but with a range of (−Ï€, Ï€].
Relationships among the inverse trigonometric functions
Complementary angles:
 \arccos x = \frac{\pi}{2}  \arcsin x
 \arccot x = \frac{\pi}{2}  \arctan x
 \arccsc x = \frac{\pi}{2}  \arcsec x
Negative arguments:
 \arcsin (x) =  \arcsin x \!
 \arccos (x) = \pi  \arccos x \!
 \arctan (x) =  \arctan x \!
 \arccot (x) = \pi  \arccot x \!
 \arcsec (x) = \pi  \arcsec x \!
 \arccsc (x) =  \arccsc x \!
Reciprocal arguments:
 \arccos x^{1} \,= \arcsec x \,
 \arcsin x^{1} \,= \arccsc x \,
 \arctan x^{1} = \tfrac{1}{2}\pi  \arctan x =\arccot x,\text{ if }x > 0 \,
 \arctan x^{1} = \tfrac{1}{2}\pi  \arctan x = \pi + \arccot x,\text{ if }x < 0 \,
 \arccot x^{1} = \tfrac{1}{2}\pi  \arccot x =\arctan x,\text{ if }x > 0 \,
 \arccot x^{1} = \tfrac{3}{2}\pi  \arccot x = \pi + \arctan x\text{ if }x < 0 \,
 \arcsec x^{1} = \arccos x \,
 \arccsc x^{1} = \arcsin x \,
If you only have a fragment of a sine table:
 \arccos x = \arcsin \sqrt{1x^2},\text{ if }0 \leq x \leq 1
 \arctan x = \arcsin \frac{x}{\sqrt{x^2+1}}
Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positive imaginary part if the square was negative real).
From the halfangle formula \tan \frac{\theta}{2} = \frac{\sin \theta}{1+\cos \theta} , we get:
 \arcsin x = 2 \arctan \frac{x}{1+\sqrt{1x^2}}
 \arccos x = 2 \arctan \frac{\sqrt{1x^2}}{1+x},\text{ if }1 < x \leq +1
 \arctan x = 2 \arctan \frac{x}{1+\sqrt{1+x^2}}
Relationships between trigonometric functions and inverse trigonometric functions
 \sin (\arccos x) = \cos(\arcsin x) = \sqrt{1x^2}
 \sin (\arctan x) = \frac{x}{\sqrt{1+x^2}}
 \cos (\arctan x) = \frac{1}{\sqrt{1+x^2}}
 \tan (\arcsin x) = \frac{x}{\sqrt{1x^2}}
 \tan (\arccos x) = \frac{\sqrt{1x^2}}{x}
General solutions
Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice in each interval of 2Ï€. Sine and cosecant begin their period at 2Ï€kâˆ’ Ï€/2 (where k is an integer), finish it at 2Ï€k + Ï€/2, and then reverse themselves over 2Ï€k + Ï€/2 to 2Ï€k + 3Ï€/2. Cosine and secant begin their period at 2Ï€k, finish it at 2Ï€k + Ï€, and then reverse themselves over 2Ï€k + Ï€ to 2Ï€k + 2Ï€. Tangent begins its period at 2Ï€kâˆ’ Ï€/2, finishes it at 2Ï€k + Ï€/2, and then repeats it (forward) over 2Ï€k + Ï€/2 to 2Ï€k + 3Ï€/2. Cotangent begins its period at 2Ï€k, finishes it at 2Ï€k + Ï€, and then repeats it (forward) over 2Ï€k + Ï€ to 2Ï€k + 2Ï€.
This periodicity is reflected in the general inverses where k is some integer:
 \sin(y) = x \ \Leftrightarrow\ y = \arcsin(x) + 2k\pi \text{ or } y = \pi  \arcsin(x) + 2k\pi
 \cos(y) = x \ \Leftrightarrow\ y = \arccos(x) + 2k\pi \text{ or } y = 2\pi  \arccos(x) + 2k\pi
 \tan(y) = x \ \Leftrightarrow\ y = \arctan(x) + k\pi
 \cot(y) = x \ \Leftrightarrow\ y = \arccot(x) + k\pi
 \sec(y) = x \ \Leftrightarrow\ y = \arcsec(x) + 2k\pi \text{ or } y = 2\pi  \arcsec (x) + 2k\pi
 \csc(y) = x \ \Leftrightarrow\ y = \arccsc(x) + 2k\pi \text{ or } y = \pi  \arccsc(x) + 2k\pi
Derivatives of inverse trigonometric functions
Simple derivatives for real and complex values of x are as follows:
\begin{align} \frac{d}{dx} \arcsin x & {}= \frac{1}{\sqrt{1x^2}}\\ \frac{d}{dx} \arccos x & {}= \frac{1}{\sqrt{1x^2}}\\ \frac{d}{dx} \arctan x & {}= \frac{1}{1+x^2}\\ \frac{d}{dx} \arccot x & {}= \frac{1}{1+x^2}\\ \frac{d}{dx} \arcsec x & {}= \frac{1}{x\,\sqrt{x^21}}\\ \frac{d}{dx} \arccsc x & {}= \frac{1}{x\,\sqrt{x^21}} \end{align} Only for real values of x:
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Answers:It's normal. I'm the same way, I just don't get math.
Answers:The standard normal distribution, which is what I think you suggest by z, has a mean of 0 and a standard deviation of 1. Therefore just put these figures (mu = 0, sigma = 1) into the general normal distribution formula that you have above and you get the formula you want.
Answers:Hi, On a TI83, press [2nd][VARS] to get to DISTR. Choose #2 normalcdf(. It needs normalcdf(lower bound, upper bound,mean,standard deviation). If you enter normalcdf(984,99999,900,30), it equals .002555 which is .2555% of the bulbs will last more than 984 hours. (99999 is just a very large number to approximate infinity.) I hope that helps!! :)
Answers:a) ANSWER: x = 1.96 Why??? NORMAL DISTRIBUTION, STANDARDIZED VARIABLE z, PROBABILITY "LOOKUP" P = 0.025 (2.5%) probability to the left of 0.250 inches Table "LOOKUP" Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 P( X <= x ) x 0.025 1.96 b) ANSWER: x = 1.51 Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 P( X <= x ) x 0.9345 1.51 c) ANSWER: x = 1.92 Inverse Cumulative Distribution Function Normal with mean = 0 and standard deviation = 1 P( X <= x ) x 0.0275 1.92
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