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# Type 1 And 2 Error Statistics

## Contents

Assume also that 90% of coins are genuine, hence 10% are counterfeit. Inserting this into the definition of conditional probability we have .09938/.11158 = .89066 = P(B|D). Examples: If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, and men with cholesterol levels over 225 are diagnosed Usually a one-tailed test of hypothesis is is used when one talks about type I error. http://u2commerce.com/type-1/type-1and-type-2-error-in-statistics.html

Applets: An applet by R. The allignment is also off a little.] Competencies: Assume that the weights of genuine coins are normally distributed with a mean of 480 grains and a standard deviation of 5 grains, The probability of a type II error is denoted by *beta*. For P(D|B) we calculate the z-score (225-300)/30 = -2.5, the relevant tail area is .9938 for the heavier people; .9938 × .1 = .09938. https://en.wikipedia.org/wiki/Type_I_and_type_II_errors

## Type 2 Error Example

what fraction of the population are predisposed and diagnosed as healthy? What is the probability that a randomly chosen coin weighs more than 475 grains and is counterfeit? P(D|A) = .0122, the probability of a type I error calculated above. What is the probability that a randomly chosen genuine coin weighs more than 475 grains?

The power of a test is (1-*beta*), the probability of choosing the alternative hypothesis when the alternative hypothesis is correct. What is the probability that a randomly chosen coin weighs more than 475 grains and is genuine? Hence P(AD)=P(D|A)P(A)=.0122 × .9 = .0110. Type 1 Error Calculator Hence P(CD)=P(C|B)P(B)=.0062 × .1 = .00062.

Examples: If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, but men predisposed to heart disease have a mean Probability Of Type 1 Error Assume 90% of the population are healthy (hence 10% predisposed). A problem requiring Bayes rule or the technique referenced above, is what is the probability that someone with a cholesterol level over 225 is predisposed to heart disease, i.e., P(B|D)=? browse this site Type II error A type II error occurs when one rejects the alternative hypothesis (fails to reject the null hypothesis) when the alternative hypothesis is true.

Type I and II error Type I error Type II error Conditional versus absolute probabilities Remarks Type I error A type I Type 1 Error Psychology Conditional and absolute probabilities It is useful to distinguish between the probability that a healthy person is dignosed as diseased, and the probability that a person is healthy and diagnosed as The former may be rephrased as given that a person is healthy, the probability that he is diagnosed as diseased; or the probability that a person is diseased, conditioned on that P(BD)=P(D|B)P(B).

## Probability Of Type 1 Error

The effect of changing a diagnostic cutoff can be simulated. If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, above what cholesterol level should you diagnose men as predisposed to heart Type 2 Error Example return to index Questions? Probability Of Type 2 Error P(D) = P(AD) + P(BD) = .0122 + .09938 = .11158 (the summands were calculated above).

This is P(BD)/P(D) by the definition of conditional probability. check my blog The latter refers to the probability that a randomly chosen person is both healthy and diagnosed as diseased. What is the probability that a randomly chosen counterfeit coin weighs more than 475 grains? A technique for solving Bayes rule problems may be useful in this context. Type 3 Error

z=(225-300)/30=-2.5 which corresponds to a tail area of .0062, which is the probability of a type II error (*beta*). Reflection: How can one address the problem of minimizing total error (Type I and Type II together)? Because the applet uses the z-score rather than the raw data, it may be confusing to you. http://u2commerce.com/type-1/type-1-and-type-2-error-statistics-examples.html The probability of a type I error is the level of significance of the test of hypothesis, and is denoted by *alpha*.

Probabilities of type I and II error refer to the conditional probabilities. Power Statistics If the cholesterol level of healthy men is normally distributed with a mean of 180 and a standard deviation of 20, at what level (in excess of 180) should men be Examples: If men predisposed to heart disease have a mean cholesterol level of 300 with a standard deviation of 30, but only men with a cholesterol level over 225 are diagnosed

## z=(225-180)/20=2.25; the corresponding tail area is .0122, which is the probability of a type I error.

One cannot evaluate the probability of a type II error when the alternative hypothesis is of the form µ > 180, but often the alternative hypothesis is a competing hypothesis of Todd Ogden also illustrates the relative magnitudes of type I and II error (and can be used to contrast one versus two tailed tests). [To interpret with our discussion of type What is the probability that a randomly chosen coin which weighs more than 475 grains is genuine? Misclassification Bias Let A designate healthy, B designate predisposed, C designate cholesterol level below 225, D designate cholesterol level above 225.

Remarks If there is a diagnostic value demarcating the choice of two means, moving it to decrease type I error will increase type II error (and vice-versa). P(C|B) = .0062, the probability of a type II error calculated above.