Stat 1040
Chapter 17 Solutions

  1. The number of aces in 180 rolls of a fair die is like the sum of 180 draws from a box with 1 ticket marked "1" and 5 tickets marked "0". The average of the box is 0.1667 and the SD of the box is 0.3727. So the expected value of the sum is
    180*0.1667 = 30,
    and the standard error of the sum is
    square root(180)*0.3727 = 5.
    In other words, we expect around 30 aces, give or take 5. The range 15 to 45 is 3 SE's each side of the EV, so we expect 99.7% of the people to get values in that range.

    1. 321/100 = 3.21.
    2. 3.78*100 = 378.
    3. The average will be between 3 and 4 if the sum is between 300 and 400. The average of the box is 3.5, and the SD is 1.7, so the expected value of the sum is 350 and the standard error is 17. The chance is about 99.7%.

  3. (a) is false and (b) and (c) are true. For case (i), the box has 12 tickets marked "2" and 26 tickets marked "-1". The average of the box is -$0.053 and the SD is $1.39. The expected value for the sum is -$53, and the standard error is $44. For case (ii), the box has 1 ticket marked "35" and 37 tickets marked "-1". The average of the box is -$0.053 and the SD is $5.76. The expected value for the sum is -$53, and the standard error is $182. Because the standard error is larger for (ii), the chance of coming out ahead is larger for (ii). So is the chance of either winning or losing more than $100. (You can check out the numbers using the normal curve).

  4. The box has 3 tickets marked "0", one ticket marked "1" and one ticket marked "3". The average of the box is 0.8 and the SD is 1.2, so the expected value of the sum is 80 and the standard error is 12. The sum will be around 80, give or take around 12 or so.