Test the foundations across one complete set

1. 1
   Mean

   Divide the total by five.

   $$\bar{x}=35/5=7$$
2. 2
   Median

   Read the third ordered observation.

   $$\widetilde{x}=5$$

Independent check: The five deviations from 7 sum to zero, and two observations lie on each side of the median 5.

1. 1
   Deviation total

   The deviations are negative 3, 0, and 3.

   $$9+0+9=18$$
2. 2
   Sample SD

   Divide by 2 and take the square root.

   $$s=\sqrt{18/2}=3$$

Independent check: Two times s² equals 2(9) = 18.

1. 1
   Complement

   Failure is not working.

   $$P(W^c)=1-P(W)$$
2. 2
   Subtract

   Complete the probability total.

   $$1-0.94=0.06$$

Independent check: The work and failure probabilities total 1.00.

1. 1
   Apply addition

   The events may overlap.

   $$P(A\cup B)=0.42+0.33-0.09$$
2. 2
   Evaluate

   Combine the three terms.

   $$P(A\cup B)=0.66$$

Independent check: A-only 0.33, both 0.09, and B-only 0.24 total 0.66.

1. 1
   Center

   Find the signed raw deviation.

   $$52-40=12$$
2. 2
   Scale

   Divide by the SD.

   $$z=12/8=1.5$$

Independent check: The inverse relation gives 40 + 1.5(8) = 52.

1. 1
   Weight each score

   Apply the stated proportions.

   $$32+54$$
2. 2
   Add

   The normalized weights already sum to one.

   $$32+54=86$$

Independent check: The result lies between 80 and 90 and is six of the ten points above 80, matching the 60 percent exam weight.

1. 1
   Find quartiles

   Use 1,2,4 below and 9,10,15 above.

   $$Q_1=2,\quad Q_3=10$$
2. 2
   Subtract

   Measure the middle-half width.

   $$\operatorname{IQR}=10-2=8$$

Independent check: The middle-half interval under the stated convention runs from 2 to 10, a width of 8.

1. 1
   Count placements

   Choose two head positions among three.

   $$\binom32=3$$
2. 2
   Multiply sequence probability

   Each length-three sequence has probability 1/8.

   $$3(1/2)^3=3/8$$

Independent check: Direct enumeration finds HHT, HTH, and THH and no other qualifying sequences.

1. 1
   Scale

   Convert SD units to raw units.

   $$0.75(16)=12$$
2. 2
   Add the mean

   The positive z-score lies above 120.

   $$x=120+12=132$$

Independent check: Forward calculation gives (132 - 120)/16 = 0.75.

1. 1
   Compare definitions

   The mean uses every magnitude; the median uses middle position.

   $$\bar{x}=\frac{\sum x_i}{n}$$
2. 2
   Choose

   An extreme magnitude can move the mean sharply without moving the middle position.

   $$\text{median is more resistant}$$

Independent check: Changing the maximum of 1,2,3,4,100 to 1,000 leaves the median 3 but changes the mean substantially.