1. In a study of 70 restaurant bills, 40 of the 70 were paid using cash. Find the posterior distribution of the binomial parameter p, the population proportion paying cash. (a) Use a beta prior distribution with a or = 2 and ,3 = 2 (b) Use a beta prior distribution with Cr = 100 and ,8 = 100. Find an approximate 95% credible interval for p without using software. 2. Suppose a corripanjtr manufactures ropes that are known to, on average, break when sup- porting 75kg. An engineer claims that her ropes are superior {have a higher break point}. but they' also cost substantially more to manufacture. In the context of a hypothesis test, we would make the following assumption about the new rope H\" : n = 75 and look for evidence of her claim Ha : n :> 75. Let's assume a is known to be 9 for the new ropes. 'We can think of that hypothesis test as a proxy,r for deciding whether to start manufacturing the new ropa. Consider the alternative value p. = Y6, which in the context of the problem would presumably not be a practically signicant departure from HD that is, given the extra cost of the ropes, the relatively small increase in average durability,r is not worth it. Also. from an individual rope standpoint {with a standard deviation of 9kg} the durability increase would not he noticeable to a consumer. (a) For an a level .01 test, compute #3 at this alternative for sample sizes n. = 100, 900+ and 2500. (b) If the observed mean of is f: = 76, what can you say about the resulting p-value when n = 2500? Is the data statistically signicant at any of the standard values of o? ((3) 'Would you want to use a sample size of 2500 along with a level or = .01 test {disre- garding the cost of such an experiment)? Explain