2.6

  1. 0
  2. 4/36
  3. 1/36

2.8

  1. No, a person could be both.
  2. [10.4% below poverty and not speaking a foreign language { ] 4.2% below poverty and speaking a foreign language [ } 16.5% not below poverty and speaking a foreign language]
  3. 10.4%
  4. 35.3%
  5. 64.7%
  6. No because P(poverty)=14.6% but P(poverty | foreign lang)= (4.2/20.7) = 20.3%, so a person is more likely to be below poverty if they speak a foreign language at home.

2.20

  1. 144/204 = 70.6%
  2. Question is slightly ambiguous. If they are asking P(self=blue & partner=blue) then 78/204 = 38.2%, but if they are asking P(partner=blue | self=blue) then 78/114 = 68.4%
  3. Same ambiguity but I will assume they are asking for conditional probability as opposed to joint.
    P(partner=blue | self=brown) = 19/54 = 35%
    P(partner=blue | self=green) = 11/36 = 30.6%
  4. No because P(self=blue) = 114/204 = 55.9% but P(self=blue | partner=blue) = 78/108 = 72.2%, so a person is more likely to have blue eyes if his partner has blue eyes.

2.30

## (a)
(13 + 15) / (13 + 15 + 59 + 8) * 59 / (27 + 59 + 8)
## [1] 0.1849944
## (b)
(13 + 59) / (13 + 15 + 59 + 8) * (12 + 15) / (12 + 15 + 59 + 8)
## [1] 0.2176932
## (c)
(13 + 59) / (13 + 15 + 59 + 8) * (13 + 15) / (13 + 15 + 59 + 8)
## [1] 0.2233795
## (d)
### Because 27 books out of 94 is not substantially different than 28 books out of 95.

2.38

## (a)
0.54 * 0 + 0.34 * 25 + 0.12 * 60  #mean in dollars
## [1] 15.7
# To be honest, I'm not sure exactly what they're looking for with the standard deviation of revenue. The formula for variance of a proportion is npq, so if you use that independently you'd get the formula below. But I believe this problem as stated is a multinomial problem (i.e., the probabilities are not independent), in which case this calculation is much more complicated and involves multiplication of the covariance matrix.
sqrt(0^2 * 0.54*(1-0.54) + 25^2 * 0.34*(1-0.34) + 60^2 * 0.12*(1-0.12))
## [1] 22.8125
## (b)
120*0.54 * 0 + 120*0.34 * 25 + 120*0.12 * 60  #mean in dollars
## [1] 1884
# see my above comment regarding the standard deviation
sqrt(0^2 * 120*0.54*(1-0.54) + 25^2 * 120*0.34*(1-0.34) + 60^2 * 120*0.12*(1-0.12))
## [1] 249.8984

2.44

  1. right skewed
  2. (21.2 + 18.3 + 15.8 + 4.7 + 2.2) = 62.2%
  3. It would be .622 * .41 = 25.5% if gender and income were independent but I would not want to make that assumption.
  4. So clearly they are not independent.