Verify multi-step statistics from a second direction

1. 1
   Recover totals

   Mean times count gives each group sum.

   $$12(8)=96,\quad8(13)=104$$
2. 2
   Combine

   Add sums and counts.

   $$\bar{x}=\frac{96+104}{12+8}=10$$

Independent check: The combined total 200 equals 20 times the reported mean 10.

1. 1
   Recover the old total

   Five observations averaging 12 total 60.

   $$\sum_{i=1}^5x_i=5(12)=60$$
2. 2
   Update

   Add 12 and divide by six.

   $$\bar{x}_{\rm new}=(60+12)/6=12$$

Independent check: The new total 72 equals six copies of the proposed mean 12.

1. 1
   Recover X's SD

   Take the nonnegative square root.

   $$\sigma_X=\sqrt9=3$$
2. 2
   Scale spread

   The shift 7 does not matter; use absolute negative 4.

   $$\sigma_Y=|-4|(3)=12$$

Independent check: The transformed variance is (-4)²(9) = 144, whose square root is 12.

1. 1
   Use the branch rule

   A joint path multiplies a marginal by the next conditional probability.

   $$P(O\cap R)=P(O)P(R\mid O)$$
2. 2
   Multiply

   Combine the branch probabilities.

   $$0.60(0.05)=0.03$$

Independent check: In 10,000 representative orders, 6,000 are online and 5 percent of those is 300, or 3 percent overall.

1. 1
   Find zero defects

   All five independent items must avoid a defect.

   $$P(X=0)=(0.98)^5=0.9039207968$$
2. 2
   Take the complement

   Every remaining count is at least one.

   $$1-0.9039207968=0.0960792032$$

Independent check: The result is slightly below the union bound 5(0.02) = 0.10 because multiple-defect outcomes would otherwise be counted repeatedly.

1. 1
   Find the lower bound

   The union cannot exceed one.

   $$P(A\cap B)\ge0.70+0.60-1=0.30$$
2. 2
   Compare

   The stated 0.20 violates that bound.

   $$0.20<0.30$$

Independent check: Substitution into the addition rule gives union 1.10, which is impossible for a probability.

1. 1
   Find the deviation

   The value is eight units below the mean.

   $$x-\mu=42-50=-8$$
2. 2
   Divide by z

   The signs cancel and SD is positive.

   $$\sigma=(-8)/(-2)=4$$

Independent check: Forward standardization gives (42 - 50)/4 = -2.

1. 1
   Read direction

   Both signs are negative, so both observations lie below their means.

   $$z_A<0,\quad z_B<0$$
2. 2
   Compare distance

   Use absolute magnitude for how far.

   $$|{-1.2}|>|{-0.8}|$$

Independent check: On a standardized number line, -1.2 is 0.4 SD farther from zero than -0.8.

1. 1
   Square s

   Sample variance is four.

   $$s^2=4$$
2. 2
   Multiply by n minus 1

   Restore the numerator of the sample variance.

   $$SS=(8-1)(4)=28$$

Independent check: Dividing 28 by 7 and taking the square root returns s = 2.

1. 1
   Find at least one

   Subtract the overlap once.

   $$P(A\cup B)=0.45+0.35-0.10=0.70$$
2. 2
   Find neither

   Neither is the complement of the union.

   $$1-0.70=0.30$$

Independent check: The four Venn regions A-only 0.35, both 0.10, B-only 0.25, and neither 0.30 sum to one.