LearningActivities ColorBlindness

From Socr

(Difference between revisions)

Revision as of 00:04, 25 October 2011

Distributome Learning Activities - Distributome Colorblindness Activity

Overview

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?

X m

~

B i n o m i a l (100, p)

.

100 women are selected at random. What is the distribution of $X f$ = the number of these women that are colorblind?

Hint: the chance that an individual woman is colorblind is

p 2

, why?

Solution:

X f

~

B i n o m i a l (100, p 2)

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?

X f

~

B i n o m i a l (n, p 2)

. Thus $Var(\frac{X_f}{n})=\frac{p^2(1-p^2)}{n}$ .

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?

Hint: The moment generating function can be used to find the fourth moment about the origin.

Hint: We want to estimate

p 2

and $\frac{X_m}{n}$ estimates

p

so it makes sense to use $(\frac{X_m}{n})^2$ as the estimator (in fact it will be the maximum likelihood estimate). We have $Var[( \frac{X_m}{n} )^2 ] = n^{-4}[E(X_m^4 ) - (E(X_m^2 ))^2 ]$ . Take

q = 1 - p

. Then the fourth moment about the origin of a binomial is

E (X 4) = n p (q - 6 p q 2 + 7 n p q - 11 n p 2 q + 6 n 2 p 2 q + n 3 p 3)

and the second moment is

E (X 2) = n p (q + n p)

. Thus $Var[( \frac{X_m}{n} )^2 ] = n^{-3}(pq + 6(n-1)p^2q^2 + 4n(n-1)p^3q)$ .

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.

Hint: Show that a normal approximation is valid for both and then compare the variances.

Solution: For large n the ratio of the variances for the estimate in part c to the estimate in part d is $\frac{Var(\frac{X_f}{n})}{Var((\frac{X_m}{n})^2 )} \sim \frac{p^2(1-p^2)}{4p^3q} = \frac{1+ p}{4p}$ . When this ratio is greater than 1, the estimator based on the sample of men will be better. Since this happens for any $p < \frac{1}{3}$ , 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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@@ Line 4: / Line 4: @@
 This [[LearningActivities| 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).
+[http://en.wikipedia.org/wiki/Color_blindness 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===

LearningActivities ColorBlindness

From Socr

Revision as of 00:04, 25 October 2011

Contents

Distributome Learning Activities - Distributome Colorblindness Activity

Overview

Goals

Hands-on Activity

Conclusions

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