Question one [20 MARKS] b). There are 250 families residing in Scandia, Pennsylvania. A random sample of 40 of these families revealed that 15 families attend church my. If a pair of six-sided fair dices is rolled once, find the event and proba- regularly, Construct the 95% confidence interval for the proportion of bility that: families attending church regularly. (i) the same number showed up on both dice. (ii) the sum of the numbers that appeared is greater than 10 or equal to 11. Question three [20 MARKS] b). Suppose you are given two opportunities to invest your savings. The first opportunity, option A, is forecast to give a profit of Ghe 1000 with a). A research center claims that 25% of college graduates think a college a probability of 0.6 and a loss of Ghe 400 with a probability of 0.4. The degree is not worth the cost. You decide to test this claim and ask a second opportunity, option B, is forecast to give a profit of Che 1200 random sample of 200 college graduates whether they think a college with a probability of 0.5 and a loss of Ghe 360 with a probabilty of 0.5. degree is not worth the cost. Of those surveyed, 21% reply yes. At Using the expected value criterion, would you choose option A or B a =0.10. is there enough evidence to reject the claim? or none? Explain your choice with the ald of a decision tree. b). The mean life of a battery used in a digital clock is 305 days. The c). A company uses three machines, labelled A, B and C for producing lives of the batteries follow the normal distribution. The battery why goods. 40%% of production goes to machine A. 50% goes to B and the recently modified to last longer. A sample of 20 modified batteries has remaining goes to C. Each machine has a different wastage rate, due a mean life of 311 days with a standard deviation of 12 days. At 5% to differing ages and manufacturing tolerances. The wastage rates are level of significance, does the modification increase the mean life of the 6% for machine A, 1 for machine B and 10% for machine C. If battery! defective items are found which machine produced them?. Use Bayes theorem and decision tree to justify. c). A medical researcher wants to determine whether there is a difference in