Apply statistics in context before calculating

1. 1
   Name the population

   The question concerns all 2,000 students.

   $$N=2000$$
2. 2
   Name the observed subset

   Only 120 provide data.

   $$n=120\Rightarrow\text{sample}$$

Independent check: The sample size n is smaller than the population size N and each surveyed student belongs to the school population.

1. 1
   Mean

   The total is eight.

   $$\mu=8/4=2$$
2. 2
   Spread

   The squared deviations total 16.

   $$\sigma=\sqrt{16/4}=2$$

Independent check: Mean-square minus square-mean is 8 - 4 = 4, whose square root is 2.

1. 1
   Restrict to commuters

   Conditioning changes the sample-space size to 20.

   $$P(W\mid C)=\frac{|W\cap C|}{|C|}$$
2. 2
   Divide

   Eight of the 20 commuters qualify.

   $$8/20=0.40$$

Independent check: Multiplying 0.40 by the commuter proportion 20/50 gives the joint proportion 8/50.

1. 1
   First red

   Three of five counters are red.

   $$P(R_1)=3/5$$
2. 2
   Then blue

   A red removal leaves both blue counters.

   $$P(B_2\mid R_1)=2/4$$
3. 3
   Multiply

   Use the ordered sequence.

   $$(3/5)(2/4)=3/10$$

Independent check: There are 20 equally likely ordered pairs of individual counters and six favorable red-blue pairs.

1. 1
   Course A

   Standardize with A's values.

   $$z_A=(90-82)/4=2$$
2. 2
   Course B

   Standardize with B's values.

   $$z_B=(76-70)/2=3$$

Independent check: B is three of its own SDs above its mean; A is only two.

1. 1
   Transform the mean

   Apply the entire linear rule.

   $$\mu_Y=5+2(10)=25$$
2. 2
   Transform spread

   Only the scale magnitude affects SD.

   $$\sigma_Y=2(3)=6$$

Independent check: A one-SD interval [7,13] transforms to [19,31], centered at 25 with half-width 6.

1. 1
   Separate A only

   Remove the shared part from A.

   $$0.60-0.30=0.30$$
2. 2
   Separate B only

   Remove the shared part from B.

   $$0.50-0.30=0.20$$
3. 3
   Add

   The only-regions are disjoint.

   $$0.30+0.20=0.50$$

Independent check: The compact formula 0.60 + 0.50 - 2(0.30) gives 0.50.

1. 1
   Choose the denominator

   The data are labeled as a sample.

   $$n-1=4$$
2. 2
   Divide

   Compute the variance before taking any square root.

   $$s^2=20/4=5$$

Independent check: The implied sample SD is √5; squaring it and multiplying by 4 restores 20.

1. 1
   Use the IQR definition

   IQR measures the Q1-to-Q3 interval.

   $$\operatorname{IQR}=Q_3-Q_1$$
2. 2
   Subtract

   Use the quartiles from the same convention.

   $$31-18=13$$

Independent check: The interval from 18 to 31 has length 13 on a number line.

1. 1
   Identify each trial

   Each trial must have two classified outcomes.

   $$\text{success or failure}$$
2. 2
   Check repetition

   Independence and one constant p make sequence probabilities comparable.

   $$P(X=k)=\binom nkp^k(1-p)^{n-k}$$

Independent check: If p changes by trial, the probability of a k-success sequence depends on which trials succeeded, so the binomial coefficient alone is insufficient.