12. The producer of a specific kind of vehicle asserts that the motors in these vehicles can run for 100,000 miles without requiring an oil change. To test this case, a buyer association intends to choose an arbitrary example of these vehicles and run them for 100,000 miles without an oil change, taking note of if any of the vehicles experience any motor difficulty during this period. At that point, they will figure a 95% certainty span for the extent of all vehicles of this kind that will experience motor difficulty inside the initial 100,000 miles. (a) Explain being 95% positive about this unique circumstance. (b) The purchaser association tests an irregular example of 5 vehicles of this kind and tracks down that 1 of the 5 vehicles experienced motor difficulty inside the initial 100,000 miles. State and check the conditions for figuring a 95% certainty stretch for the extent of all vehicles of this kind that would experience motor difficulty inside the initial 100,000 miles. Try not to figure the stretch. To examine what happens while ascertaining a certainty span for an extent with a little example size, 1000 recreated irregular examples of 5 vehicles were chosen from a populace of vehicles where 30% would experience motor difficulty inside the initial 100,000 miles. The table sums up the appropriation of p? = the extent of vehicles in the example with motor difficulty for these 1000 reproduced tests. For every conceivable worth of p? , the table additionally shows the lower and upper endpoints of a 95% certainty span for p utilizing the recipe: p? ? 1 ? p? ? 5 For instance, in 309 of the 1000 reenacted tests, the worth of p? was 0.4 with a comparing certainty time period 0.029 to 0.829. p? ?1.96 p? = extent of vehicles in the example with motor difficulty Numbe r of test s Lower 95% certainty limit Upper 95% certainty limit 0 168 0 0.2 360 - 0.151 0.551 0.4 309 - 0.029 0.829 0.6 133 0.171 1.029 0.8 28 0.449 1.151 1 2 1 5 (c) According to the reproduction, how frequently will these 95% certainty stretches catch the genuine extent of vehicles of this kind that will experience motor difficulty in the initial 100,000 miles? Clarify how you decided your answer. (d) When the example size is little, an elective technique for figuring a 95% certainty span for an extent includes adding 2 victories and 2 disappointments to the example, ascertaining the worth of p? utilizing the first example in addition to these 4 extra perceptions, and utilizing the equation: p? ? 1 ? p? ? p? ?1.96 (I) In the customer association's example of 5 vehicles, 1 experienced motor difficulty in the initial 100,000 miles. What is the n worth of p? that ought to be utilized to ascertain the certainty span for this example, accepting that the elective technique is to be utilized? (ii) The table underneath shows the aftereffects of the reenactment from part (c), alongside the new lower and upper 95% certainty stretch limits utilizing the elective strategy. p? = extent of vehicles in the example with motor difficulty Numbe r of test s Alternative lower 95% certainty limit Alternative upper 95% certainty limit 0 168 - 0.049 0.494 0.2 360 0.025 0.641 0.4 309 0.120 0.769 0.6 133 0.231 0.880 0.8 28 0.359 0.975 1 2 0.506 1.049 According to the reenactment, shows improvement over the customary technique in this specific situation? Clarify your thinking

Example 1.4 Ehrenfest Chain Model for exchange of heat or gas molecules between two bodies. Was a way to explain the second law of thermodynamics. Basic idea is, there are d molecules and two bodies in a closed system. The interest is in body 1, which has r molecules. The loss of a molecule is a function of the number of molecules so that Pr(lose a molecule in body 1) = = =1 -Pr(gain. ..). r molecules dor molecules wanttion probabilides when d =3 1 word +1 wp 1-/d body 1 body 2 Let X, = the number of molecules at exchange (time) n, then S = {0, 1,...,d} and transition probabilities r/d if k = r - 1 P(r,k) =