Problem 1 (Deliveries) A food delivery company prides itself for being able to deliver food to your door with 30 minutes of the time the order was placed. An internal audit wants to check whether the company can actually comply with its pledge. For this purpose, they collect data on 25 deliveries and estimate the average delivery time of 33 minutes. Assuming that the standard deviation of the delivery times is 10 minutes, you need to decide whether the company on average fulfils its pledge of under 30 minutes delivery or not. a) Define your null and the alternative hypotheses. (10 points) b) Is it a one- or two-sided test? Justify. (10 points) Calculate the test statistic. (10 points) d) Calculate the p-value for this test and interpret. (10 points) e) Can you reject the null hypothesis at 5% significance level? Justify (10 points) f) Find a threshold delivery time above which the company can no longer claim that it delivers the food at your door within 30 minutes, assuming you are willing to commit the type I error with probability of 1%. (10 points) g) What is the probability of detecting the actual delivery time of about 40 minutes? Calculate the probability of type II error and the power of the test assuming the significance level of 1%. (10 points)