1. It is important for airlines to follow the published scheduled depar- ture times of flights. Suppose that one airline that recently sampled the records of 246 flights originating in London found that 20 flights were delayed for severe weather; 4 flights were delayed for maintenance concerns, and all the other flights were on time. (a) Compute a 95% confidence interval for the percentage of on-time departures. (b) Compute a 95% confidence interval for the occurrence of severe weather.2. Suppose you have a random sample of observations X1, ...,Xn of the number of seconds of advertisements shown before movie screenings at the cinema. You use the sample mean X to estimate the population mean u. (a) Suppose you know that the distribution of X1- is normal with known variance 02. Describe (using math) how to construct a 95% condence interval for u that is symmetric around X. (b) Now suppose 02 is unknown. Describe how to construct the same kind of 95% condence interval as in part (a) now. How does your condence interval differ when 02 is unknown. (c) Suppose now that you are unsure and are unwilling to assume that the distribution of the Xi is normal. How can you construct a symmetric 95% condence interval now? Is the interpretation any different than in parts (a) and (b)? 3. (Empirical Exercise) You need to use R to address the following em- pirical questions. Use the inbuilt dataset "USArrests" by typing > data("USArrests") in your R console. In your variable space, now you have a data frame "USArrests" which contains statistics about violent crime rates by US state. (a) Compute means and variances of " Murder", " Assault", "Urban- Pop", and "Rape". (b) Plot "Murder" and " Assault" in a scatter plot. Explain the rela- tionship between the two crimes. (c) Compute covarance and correlation coefficient of "Murder" and "Assault". Elaborate your answer in (b). (d) Plot "Murder" and other 2 variables (" UrbanPop" and "Rape"). Compute covariance and correlation coefficients of them. Explain the relationships between "Murder" and other variables. (e) State a hypothesis on the relationships between crime types that you want to test. How can you test your hypothesis using statis- tics