1. **Problem statement:** We have a normally distributed random variable $X$ with mean $\mu = 10$ and standard deviation $\sigma = 2$. (a) Find the probability that $X$ is more than $1.5$ standard deviations above the mean. (b) Given that the probability that $X$ is more than $k$ standard deviations above the mean is $0.1$, find the value of $k$. --- 2. **Recall the standard normal distribution:** If $X \sim N(\mu, \sigma^2)$, then the standardized variable $$Z = \frac{X - \mu}{\sigma}$$ is standard normal distributed $Z \sim N(0,1)$. 3. **Part (a):** We want $P\left(X > \mu + 1.5\sigma\right) = P\left(Z > 1.5\right)$. Using standard normal tables or a calculator, $$P(Z > 1.5) = 1 - P(Z \leq 1.5) = 1 - 0.9332 = 0.0668.$$ So, the probability that $X$ is more than $1.5$ standard deviations above the mean is approximately $0.0668$. 4. **Part (b):** We are given: $$P\left(X > \mu + k\sigma\right) = P(Z > k) = 0.1.$$ We want to find $k$ such that the upper tail probability is $0.1$. From standard normal distribution tables or using the inverse CDF (quantile function), $$k = z_{0.9}$$ where $z_{0.9}$ is the 90th percentile of the standard normal distribution. Looking up or calculating, $$k \approx 1.2816.$$ Thus, the value of $k$ is approximately $1.28$. --- **Final answers:** (a) $P\left(X > 10 + 1.5 \times 2\right) = P(Z > 1.5) \approx 0.0668$ (b) $k \approx 1.28$ such that $P(Z > k) = 0.1$