1. **Problem:** Find the probability that $z$ is between 0.43 and 0.72 for a standard normal distribution. 2. **Formula and rules:** For a standard normal variable $z$, the probability between two values $a$ and $b$ is given by: $$P(a < z < b) = \Phi(b) - \Phi(a)$$ where $\Phi(x)$ is the cumulative distribution function (CDF) of the standard normal distribution. 3. **Step-by-step solution:** - Find $\Phi(0.72)$ and $\Phi(0.43)$ from standard normal tables or a calculator. - From tables or calculator: $$\Phi(0.72) \approx 0.7642$$ $$\Phi(0.43) \approx 0.6664$$ - Calculate the probability: $$P(0.43 < z < 0.72) = 0.7642 - 0.6664 = 0.0978$$ **Final answer:** $$\boxed{0.0978}$$ --- 2a. **Problem:** Find $P(-2 < z < 1)$. - Using CDF values: $$\Phi(1) \approx 0.8413$$ $$\Phi(-2) \approx 0.0228$$ - Calculate: $$P(-2 < z < 1) = 0.8413 - 0.0228 = 0.8185$$ 2b. **Problem:** Find $P(z > 3)$. - Using CDF: $$\Phi(3) \approx 0.9987$$ - Calculate: $$P(z > 3) = 1 - 0.9987 = 0.0013$$ 2c. **Problem:** Find $P(z < -2)$. - Using CDF: $$\Phi(-2) \approx 0.0228$$ - So: $$P(z < -2) = 0.0228$$ 2d. **Problem:** Find $P(z > -1.5)$. - Using CDF: $$\Phi(-1.5) \approx 0.0668$$ - Calculate: $$P(z > -1.5) = 1 - 0.0668 = 0.9332$$ 2e. **Problem:** Find $P(z = -1)$. - For continuous distributions, the probability at a single point is zero: $$P(z = -1) = 0$$ --- **Summary of answers:** - $P(0.43 < z < 0.72) = 0.0978$ - $P(-2 < z < 1) = 0.8185$ - $P(z > 3) = 0.0013$ - $P(z < -2) = 0.0228$ - $P(z > -1.5) = 0.9332$ - $P(z = -1) = 0$