Problem ll! [3 points]. You work in a store where the chance of each customer making a purchase is p. Using Chebyshev's inequality, you want to estimate the value of p within 5%. How many randomly and independently selected customers do you need to observe to have 90% condence in your estimate of p? Problem 1] [9 points]. Suppose that one would like to test the hypothesis that at least 10% of students suffer from allergies. One collects a random sample cf225 students and 21 of them suffer from allergies. a. State the null and alternative hypotheses. h. Obtain a test statistic and a pvalue. e. State the conclusion at the a = {1.05 level. Problem 12 [8 points]. X is a random variable defined by its probability mass function: Px (k; 0) = 0(1 - 0)k-1 for k = 1,2, 3, ... One needs to decide between two hypotheses for the parameter 0. Specifically, the null hypothesis is that 0 is 0.50 while the alternative hypothesis is that 0 is 0.10. a. Design a test to decide between the two hypotheses at a significance level of 0.05 b. Find the probability for a type-II error for your test