1. Given $X \sim N(44, 1^2)$, find the probabilities: 1.1 Calculate $P(X \leq 44.4)$: - Mean $\mu = 44$, standard deviation $\sigma=1$. - Find z-score: $z = \frac{44.4-44}{1} = 0.4$. - Look up $P(Z \leq 0.4)$ in standard normal table or use a calculator: $P(Z \leq 0.4) \approx 0.6554$. 1.2 Calculate $P(X > 45.2)$: - $z = \frac{45.2 - 44}{1} = 1.2$. - Find $P(Z > 1.2) = 1 - P(Z \leq 1.2) \approx 1 - 0.8849 = 0.1151$. 1.3 Calculate $P(X > 41.5)$: - $z = \frac{41.5 - 44}{1} = -2.5$. - $P(X > 41.5) = P(Z > -2.5) = 1 - P(Z \leq -2.5)$. - From tables, $P(Z \leq -2.5) \approx 0.0062$. - So $P(X > 41.5) = 1 - 0.0062 = 0.9938$. 1.4 Calculate $P(X < 43.2)$: - $z = \frac{43.2 - 44}{1} = -0.8$. - $P(X < 43.2) = P(Z < -0.8) \approx 0.2119$. 2. Given $X \sim N(69, 15^2)$, find the probabilities: 2.1 Calculate $P(X \leq 78)$: - $z = \frac{78 - 69}{15} = 0.6$. - $P(Z \leq 0.6) \approx 0.7257$. 2.2 Calculate $P(X \leq 49.5)$: - $z = \frac{49.5 - 69}{15} = -1.3$. - $P(Z \leq -1.3) \approx 0.0968$. 2.3 Calculate $P(X > 70.5)$: - $z = \frac{70.5 - 69}{15} = 0.1$. - $P(Z > 0.1) = 1 - P(Z \leq 0.1) = 1 - 0.5398 = 0.4602$. 2.4 Calculate $P(55.5 < X < 73.5)$: - Convert to z-scores: $z_1 = \frac{55.5 - 69}{15} = -0.9$, $z_2 = \frac{73.5 - 69}{15} = 0.3$. - $P(55.5 < X < 73.5) = P(-0.9 < Z < 0.3) = P(Z < 0.3) - P(Z < -0.9)$. - From tables, $P(Z < 0.3) \approx 0.6179$, $P(Z < -0.9) \approx 0.1841$. - So, $0.6179 - 0.1841 = 0.4338$. Final answers: $P(X \leq 44.4) \approx 0.6554$ $P(X > 45.2) \approx 0.1151$ $P(X > 41.5) \approx 0.9938$ $P(X < 43.2) \approx 0.2119$ $P(X \leq 78) \approx 0.7257$ $P(X \leq 49.5) \approx 0.0968$ $P(X > 70.5) \approx 0.4602$ $P(55.5 < X < 73.5) \approx 0.4338$