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Colorblindness is any abnormality of the color vision system that causes a person to see colors differently than most people, or to have difficulty distinguishing among certain colors (www.visionrx.com).Colorblindness is gender-based, while the majority of sufferers being males.Roughly 8% of white males have some form of colorblindness, while the incidence among white females is only 1%. A random sample of 20 white males and 40 white females was chosen.Let X be the number of males (out of the 20) who are colorblind.Let Y be the number of females (out of the 40) who are colorblind.Let Z be the total number of colorblind individuals in the sample (males and females together).Which of the following is true about the random variables X, Y, and Z? Check all that apply. X is binomial with n = 20 and p = .08. Y is binomial with n = 40 and p = .01. Z is not binomial.

Question

Colorblindness is any abnormality of the color vision system that causes a person to see colors differently than most people, or to have difficulty distinguishing among certain colors (www.visionrx.com).Colorblindness is gender-based, while the majority of sufferers being males.Roughly 8% of white males have some form of colorblindness, while the incidence among white females is only 1%. A random sample of 20 white males and 40 white females was chosen.Let X be the number of males (out of the 20) who are colorblind.Let Y be the number of females (out of the 40) who are colorblind.Let Z be the total number of colorblind individuals in the sample (males and females together).Which of the following is true about the random variables X, Y, and Z? Check all that apply. X is binomial with n = 20 and p = .08. Y is binomial with n = 40 and p = .01. Z is not binomial.

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Solution

The statement "X is binomial with n = 20 and p = .08" is true. This is because X represents the number of colorblind males out of a sample of 20, and we know that the probability of a male being colorblind is 0.08. This fits the definition of a binomial distribution, which is used when there are exactly two mutually exclusive outcomes of a trial (in this case, a male being colorblind or not).

The statement "Y is binomial with n = 40 and p = .01" is also true. Y represents the number of colorblind females out of a sample of 40, and the probability of a female being colorblind is 0.01. This also fits the definition of a binomial distribution.

The statement "Z is not binomial" is true. Z represents the total number of colorblind individuals in the sample, both males and females. The probabilities of a male and a female being colorblind are not the same, so the outcomes are not mutually exclusive, which is a requirement for a binomial distribution. Therefore, Z does not follow a binomial distribution.

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