1. **Problem statement:** We have two normal random variables $X \sim N(7,a^2)$ and $Y \sim N(19,a^2)$ with $a>0$. (a) Find $b$ such that $P(X>b) = P(Y>22)$. (b) Find approximate value of $P(7 - a < X < 7 + a)$ to two significant figures. (c) Given $a=3$, find approximate value of $P(Y<22)$ to two significant figures. --- 2. **Recall properties of normal distribution:** - If $Z \sim N(0,1)$ (standard normal), then for any $X \sim N(\mu, \sigma^2)$, we have $P(X > x) = P\left(Z > \frac{x-\mu}{\sigma}\right)$. - The symmetry and standard normal tables or calculator can be used to find probabilities. --- 3. **Part (a): Find $b$ such that $P(X>b) = P(Y>22)$** - Since $X \sim N(7,a^2)$ and $Y \sim N(19,a^2)$, standardize: $$P(X>b) = P\left(Z > \frac{b-7}{a}\right)$$ $$P(Y>22) = P\left(Z > \frac{22-19}{a}\right) = P\left(Z > \frac{3}{a}\right)$$ - Set equal: $$P\left(Z > \frac{b-7}{a}\right) = P\left(Z > \frac{3}{a}\right)$$ - Since $P(Z>z)$ is decreasing in $z$, this implies: $$\frac{b-7}{a} = \frac{3}{a}$$ - Multiply both sides by $a$ (positive, so inequality direction preserved): $$b - 7 = 3$$ - Solve for $b$: $$b = 10$$ --- 4. **Part (b): Approximate $P(7 - a < X < 7 + a)$** - Standardize the interval: $$P(7 - a < X < 7 + a) = P\left(-1 < \frac{X-7}{a} < 1\right) = P(-1 < Z < 1)$$ - From standard normal tables, $P(-1 < Z < 1) \approx 0.68$ (68%) - Rounded to two significant figures: $0.68$ --- 5. **Part (c): Given $a=3$, find $P(Y < 22)$** - Standardize: $$P(Y < 22) = P\left(Z < \frac{22 - 19}{3}\right) = P(Z < 1)$$ - From standard normal tables, $P(Z < 1) \approx 0.8413$ - Rounded to two significant figures: $0.84$ --- **Final answers:** - (a) $b = 10$ - (b) $P(7 - a < X < 7 + a) \approx 0.68$ - (c) $P(Y < 22) \approx 0.84$