1. **Problem statement:** Chris sees an average of 15 meteors per hour on Monday night. We want to find probabilities for meteors seen in shorter intervals and perform a hypothesis test for Friday night data. 2. **Poisson distribution formula:** The number of meteors seen in a time interval follows a Poisson distribution with parameter $\lambda = \text{rate} \times \text{time}$. 3. **Calculate $\lambda$ for Monday night:** - Rate per hour = 15 meteors - For 20 minutes = $\frac{20}{60} = \frac{1}{3}$ hour - So, $\lambda = 15 \times \frac{1}{3} = 5$ 4. **(a)(i) Probability of at least 6 meteors in 20 minutes:** - $P(X \geq 6) = 1 - P(X \leq 5)$ - Use Poisson CDF for $k=5$ with $\lambda=5$ 5. **(a)(ii) Probability of no more than 3 meteors in 20 minutes:** - $P(X \leq 3)$ with $\lambda=5$ 6. **(b) Hypotheses for Friday night test:** - Null hypothesis $H_0$: $\lambda = 15$ meteors per hour (no increase) - Alternative hypothesis $H_a$: $\lambda > 15$ meteors per hour (increase) 7. **(c) Critical region for 30 minutes observation:** - Rate per 30 minutes under $H_0$: $\lambda_0 = 15 \times 0.5 = 7.5$ - Significance level $\alpha = 0.05$ - Find smallest integer $k$ such that $P(X \geq k) \leq 0.05$ for $X \sim \text{Poisson}(7.5)$ 8. **(d) Conclusion with observed 12 meteors in 30 minutes:** - If 12 is in critical region, reject $H_0$; else fail to reject - Reason based on probability of observing 12 or more meteors under $H_0$ --- **Calculations:** - For (a)(i): $$P(X \geq 6) = 1 - \sum_{k=0}^5 \frac{5^k e^{-5}}{k!}$$ - For (a)(ii): $$P(X \leq 3) = \sum_{k=0}^3 \frac{5^k e^{-5}}{k!}$$ - For (c): Find $k$ such that $$P(X \geq k) = 1 - P(X \leq k-1) \leq 0.05$$ where $X \sim \text{Poisson}(7.5)$. Using Poisson tables or software: - $P(X \leq 12)$ for $\lambda=7.5$ is approximately 0.989 - $P(X \geq 13) = 1 - 0.989 = 0.011 < 0.05$ - $P(X \geq 12) = 1 - P(X \leq 11)$, and $P(X \leq 11)$ is about 0.963, so $P(X \geq 12) = 0.037 < 0.05$ Thus, critical region is $X \geq 12$. - Observed $X=12$ is in critical region, so reject $H_0$. --- **Final answers:** 1. (a)(i) $P(X \geq 6) = 1 - P(X \leq 5)$ with $\lambda=5$. 2. (a)(ii) $P(X \leq 3)$ with $\lambda=5$. 3. (b) $H_0: \lambda=15$, $H_a: \lambda>15$. 4. (c) Critical region: $X \geq 12$ meteors in 30 minutes, with probability $\leq 0.05$ under $H_0$. 5. (d) Since 12 meteors observed is in critical region, reject $H_0$. There is evidence at 5% level that meteor rate increased on Friday night.