In total 231 boxes were collected and opened, in which they found 130 white balls, 3 red balls, and 5 black balls. The remaining boxes were empty. a) Construct a 97 % confidence interval for the probability of finding a ball in a box. b) According to one claim that only 80 % of boxes contain a ball. When a box is not empty, it contains a red ball in 2 % of cases and that a black ball in 3 % of cases. Is the claim about black balls correct? Verify it at 5 % significance level by a one-tailed test. c) What the claim surprisingly did not realize, is that in fact there are three different types of boxes, one possibly containing a white ball, the other containing red balls and the third type containing black ball, and that these three types of balls have clearly distinctive shape so that they can be easily identified. In fact, in the sample of 231 boxes, there were 30 boxes that could contain a red ball and 31 boxes that could contain a black ball. Construct a 95 % confidence interval that allows you to test the claim that the probability of finding a red ball in a red boxes is equal to the probability of finding a black ball in a black boxes. Solely on the basis of this confidence interval, state the outcome of this hypotheses test. For this assignment, assume the independence of probabilities. Do you need any other assumptions?