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Practice with Contingency Tables and Conditional Probability
Show your work or explain your thinking. There may be a few different methods that work.
1. According to the 2019 U.S. Census, the country's total population was 328,239,523 people. The Census only uses male and female genders: 50.8% of the population was female, and the rest male. The census does not ask about handedness, but other studies estimate 11.8% of males are left-handed, and 9.6% of females are left-handed. Use this information to fill out the table below.
For the follwing problems, use the following labels for situations: L = left handed, R = right handed, M = male, and F = female.
2. The statement "11.8% of males are left-handed" can be rephrased as the probability a person is left-handed, given that the person is male, is 11.8%. The notation for this conditional probability is P(L|M) and we could write the equation P(L|M) = 0.118. Write a similar equation for the statement "9.6% of females are left-handed".
3. Would you say that gender and handedness are independent? That is, does one’s gender affect one’s likelihood of being left handed or right handed?
4. Recall that for independent events A and B, P(A and B) = P(A) × P(B). Can we say that P(M and L) = P(M) × P(L)?
5. In general, if events A and B are not independent, then we write P(A and B) = P(A) × P(B|A). Knowing L and M are not independent, write P(M and L) as a product.
6. Knowing L and F are not independent, write P(F and L) as a product.
The DSM-5 estimates that 0.005% to 0.014% of male newborns will experience gender dysphoria at some time in their life. Studies have linked high levels of maternal first-trimester intrauterine testosterone to both left-handedness and gender dysphoria. Furthermore, a 2001 study found 19.5% of its boys with gender dysphoria were left-handed.
7. Use this information to finish filling out the table below for Oregon's population of 700,000 boys of age 14 years or younger, picking the high estimate of 0.014% and assuming that 11.8% of boys without gender dysphoria are left-handed.
|Boys with Gender Dysphoria||Boys without Gender Dysphoria||Total|
|Left Handed||Step 3 is 0.118 × 699,902 =|
|Total||Step 1 is 0.00014 × 700,000 =|
|Step 2 is 700,000 − 98 =|
For the follwing problems, use the following labels for situations: L = left handed, R = right handed, N = boy without Gender Dysphoria, and Y = boy with Gender Dysphoria.
8. Find P(L|N).
9. Find P(N|L).
10. Find P(N|R).
11. Find P(Y|L).
12. Find P(Y|R).
13. Find P(Y|L) − P(Y|R). (This is an absolute change.)
14. Divide the answer to #13 by P(Y|R). (This is a relative change.)
These problems are not graded. They are only to help you practice with our math topics. Do not rush to look at answers! First ask for hints from your instructor or classmates. But if you are really ready, the answers are on a spreadsheet this time, here.