AP Statistics 4.9 Combining Random Variables- MCQs - Exam Style Questions
▶️ Answer/Explanation
Let \(S\sim \mathcal N(225,25)\) and \(E\sim \mathcal N(240,15)\), independent. Consider \(S-E\).
Then \(S-E \sim \mathcal N\!\big(\mu_S-\mu_E,\ \sqrt{\sigma_S^2+\sigma_E^2}\big) = \mathcal N\!\big(-15,\ \sqrt{25^2+15^2}\big) = \mathcal N\!\big(-15,\ \sqrt{850}\big).\)
We want \(P(S>E)=P(S-E>0)\). Standardize: \[ z=\frac{0-(-15)}{\sqrt{850}}=\frac{15}{\sqrt{850}}\approx \frac{15}{29.154}\approx 0.515. \] Thus \(P(S>E)=P(Z>0.515)=1-\Phi(0.515)\approx 1-0.697\approx 0.303.\)
✅ Answer: (D)
Why others are wrong (common mistakes):
(A), (B), (C), (E) come from using an incorrect standard deviation (e.g., subtracting or adding \(25\) and \(15\), or using one of them alone) rather than \(\sqrt{25^2+15^2}\).
▶️ Answer/Explanation
Let \(X\) = time before training, \(Y\) = time after training for the same student, so \(X\) and \(Y\) are dependent (paired).
Mean of a difference: \(\mu_{X-Y}=\mu_X-\mu_Y=402-368=34\) seconds.
Standard deviation of a difference: \(\sigma_{X-Y}=\sqrt{\sigma_X^2+\sigma_Y^2-2\operatorname{Cov}(X,Y)}\).
Because the covariance \(\operatorname{Cov}(X,Y)\) is unknown (and not \(0\) since the variables are not independent), \(\sigma_{X-Y}\) cannot be determined from the given information.
(If \(X\) and \(Y\) were independent, one would get \(\sqrt{40^2+30^2}=50\) seconds, but independence is false here.)
✅ Answer: (D)