For all the problems in this assignment, you SHO ULD present your solution step by step instead of giving the answers only. Question 1: (151m) Assume identical twins are always of the same sex, equally likely boys or girls. Assume that for fraternal twins the rstborn is equally likely to be a boy or a girl, and so is the m independently of the first. Assume that proportion 33 of twins are identical, and proportion q = 1 p are fraternal. Find formula in terms of p for the following probabilities for twins: a) beoth boys] b) PUirstborn = boy and secondborn = girl) c) PUirstborn = boy} d) PUirstborn = girl) Question 2: (Nuts) Alice is studying the gender distribution of left-handers and right-handers in and; After a survey, she compiled the following table: Question: a} Which gender is more likely to be lefthanded. male or female? Calculate the probability that a male is left-handed. i.e.. mm. and the probability that a female is lefthanded. i.e.. E_i,_left lfemale). b) Use Bayes' Formula to calculate the probability MW, and the prob ability MW based on the answer of question (a). If a student is left-handed, is the student more likely to be a female or male? Justify your answer. Question 3: (lpts) A manufacturing process produces integrated circuit chips, in which the fraction of bad chips is around 20%. Thoroughly testing a chip to determine whether it is good or bad is rather expensive, so a cheap test is tried. 99% of the good chips will pass the cheap test, but 10% of the bad chips will also pass it. a) Given a chip passes the cheap test, what is the probability that it is a good chip? Use at least 3 decimal places. b) In a batch of integrated circuit chips where all chips pass the cheap test, what is the percentage of bad chips? Question 4: (ipts) To test for COVID-19, rapid tests (antigen tests] are quicker (give results within 30 minutes) but they are not accurate. Suppose the fraction of the population being infected by COVID-19 is 1%. A rapid test is available to test for the virus. W a healthy person, the chance of being falsely diagnosed as having the virus is 2.5%. For someone with the virus, the chance of being falsely diagnosed as healthy is 6.7%. Suppose the test is performed on a person selected at random from the population. a) What is the probability that the test shows a positive result (meaning the person is diagnosed as positive, either correctly or incorrectly)? b) Suppose the test shows a positive result, what is the probability that the person tested actually has the virus? c) Suppose the test shows a negative result, what is the probability that the person tested actually has the virus? Question 5: (151m) The table below shows some past sales data from a clothing company. Each row shows the sales for a year, and the amount s . ent on advertisin_ in that ear. Advertisin_ (Million Dollars) Sales (Million Dollars) a) Use linear regression to calculate the relationship between the advertising and sales. b] Calculate the correlation coefficient between the amounts of advertising and sales. c) If there is no advertising, what is the expected sales? If the amount of advertising is 63 million dollars in the next w is the prediction of the sales? Question 6: (lpts) Suppose that in a particular application requiring a single battery, the mean lifetime of the battery is 4 weeks, with a stande deviation of 1 week. The battery is replaced by a new one when it dies, and so on. Assume the lifetime of each battery is independent. What, approximately, is the probability that more than 14 replacements will have to be made in a two-year period, starting at the time of installation of a new battery? [Hint Normal approximation. For each year we assume that there are 52 weeks. Assume X; is the lifetime in weeks of the 1,1,]; battery, and S= ererXstudgo is approximately normally distributed. As such, the question turns out to be finding the probability that S