AP Statistics 2.9 Parameters of Random Variables Questions

Model Solution
$$\begin{aligned} E(X)&=1(0.12)+2(0.22)+3(0.28)\\ &\quad+4(0.22)+5(0.14)+6(0.02)\\ &=3.10\text{ bedrooms}. \end{aligned}$$

Scoring components
3. Calculates the mean 3.10.
4. Shows supporting work or justification.

Scoring

Parts (a), (b), (c), and (d) are scored as essentially correct (E), partially correct (P), or incorrect (I).
Part (a) is scored as follows:
Essentially correct (E) if the probability is computed correctly with work shown.
Partially correct (P) if the correct answer is given, but no work is shown;
OR
if appropriate work is shown but the answer is incorrect or missing;
OR
if one of the incorrect cumulative probabilities $P(X<1), P(X \leq 1)$, or P(X>1) is stated AND computed correctly.

Incorrect (I) if the response does not meet the criteria for E or P.
Note: The probability can be calculated as 1-P(X=0)=1-0.15=0.85.
Part (b) is scored as follows:
Essentially correct (E) if the expected value is computed correctly with work shown.
Partially correct (P) if the correct answer is given, but no work is shown;
OR
if appropriate work is shown but the answer is incorrect or missing.
Incorrect (I) if the response does not meet the criteria for E or P.
Part (c) is scored as follows:
Essentially correct (E) if the probability is computed correctly, with work shown that includes correct numerical values for both the numerator and denominator.

Partially correct (P) if the response includes a numerator and denominator in calculating the conditional probability, with one (numerator or denominator) correct in numerical value and the other incorrect;

OR
if the correct answer is given, but no work is shown.
Incorrect (I) if the response does not meet the criteria for E or P.
Part (d) is scored as follows:
Essentially correct (E) if the response provides the correct answer (greater than) with a reasonable explanation based on the fact that with X=0 eliminated, the probabilities for X=1, X=2, and X=3 all increase;

OR
if the response provides the correct answer (greater than) with a reasonable explanation based on the fact that with X=0 eliminated, the balance point of the distribution increases;

OR
if the response provides the correct answer (greater than) and the conditional expected value is computed correctly with work shown.

Partially correct (P) if the response provides the correct answer (greater than) with a weak explanation, such as "Yes, because 0 is eliminated";

OR
if the response provides the correct answer (greater than) with a reasonable but incorrect attempt to calculate the conditional expected value using a revised probability distribution with only the values X=1, X=2, and X=3, and corresponding probabilities greater than or equal to those in the original probability distribution.

Incorrect (I) if the response does not provide the correct answer (greater than);
OR
if the response provides the correct answer (greater than) with an incorrect explanation or no explanation.

Note: The conditional expected value is:
$$1\left(\frac{0.21}{0.85}\right)+2\left(\frac{0.40}{0.85}\right)+3\left(\frac{0.24}{0.85}\right) \approx 1(0.247)+2(0.471)+3(0.282)=2.04 \text { machines. }$$

Each essentially correct (E) part counts as 1 point. Each partially correct (P) part counts as $\frac{1}{2}$ point.
4 Complete Response
3 Substantial Response
2 Developing Response
1 Minimal Response

If a response is between two scores (for example, $2 \frac{1}{2}$ points), use a holistic approach to decide whether to score up or down, depending on the overall strength of the response and communication.