LearningActivities ColorBlindness

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Distributome Learning Activities - Distributome Colorblindness Activity

Overview

Colorblindness - Can you see the number in this image?

This Distributome Activity illustrates an application of probability theory to study Colorblindness.

Colorblindness results from an abnormality on the X chromosome. The condition is thus rarer in women since a woman would need to have the abnormality on both of her X chromosomes in order to be colorblind (whether a woman has the abnormality on one X chromosome is essentially independent of having it on the other).

Goals

The goal of this activity is to demonstrate an efficient protocol of estimating the probability that a randomly chosen individual may be colorblind.

Hands-on Activity

Suppose that p is the probability that a randomly selected man is colorblind.

• 100 men are selected at random. What is the distribution of X_m = the number of these men that are colorblind?

• 100 women are selected at random. What is the distribution of X_f = the number of these women that are colorblind?

• To estimate the probability that a randomly selected woman is colorblind, you might use the proportion of colorblind women in a sample of n women. What is the variance of this estimator?

See the Answer

  X_f~Binomial(n,p²). Thus .

• Alternatively, to estimate the probability that a randomly selected woman is colorblind, you might use the square of the proportion of colorblind men in a sample of n men. Explain why this estimate makes sense. What is the variance of this estimator?

See the Answer

  We want to estimate p² and estimates p so it makes sense to use as the estimator (in fact it will be the maximum likelihood estimate). We have . Take q = 1 − p. Then the fourth moment about the origin of a binomial is E(X⁴) = np(q − 6pq² + 7npq − 11np²q + 6n²p²q + n³p³) and the second moment is E(X²) = np(q + np). Thus .

1. For large samples, is it better to use a sample of men or a sample of women to estimate the probability that a randomly selected women is colorblind? Explain.

{{hidden| See a Hint| Show that a normal approximation is valid for both and then compare the variances.}

See the Answer

  For large n the ratio of the variances for the estimate in part c to the estimate in part d is . When this ratio is greater than 1, the estimator based on the sample of men will be better. Since this happens for any , which is clearly the case for colorblindness, it is better to use a sample of men to estimate the probability that a random woman is colorblind.

Conclusions

You can also use the delta method to find the approximate variance for the estimator above.

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