Prob 1.The carbon content in a block of steel specimens are sampled. 8 samples have the following weight of carbon: 7.3, 8.6, 10.4, 16.1, 12.2, 15.1, 14.5, and 9.3milligrams. Calculate:(a) the mean(b) the variance.Prob 2.The length of 1000 steel bars are approximately normally distributed with a mean of 174.5 centimeters and a standard deviation of 6.9 centimeters. Suppose 200random samples of size 25 are drawn from this population and the means recorded to the nearest tenth of a centimeter. Determine(a) the mean and standard deviation of the sampling distribution of X(b) the number of sample means that fall between 172.5 and 175.8 centimeters inclusive(c) the number of sample means falling below 172.0 centimeters.Prob 3.The average life of a car tire is 7 years, with a standard deviation of 1 year. Assuming that the lives of these tires follow approximately a normal distribution,find(a) the probability that the mean life of a random sample of 9 such tires falls between 6.4 and 7.2 years(b) the value of x to the right of which 15% of the means computed from random samples of size 9 would fall.Prob 4.Two aluminum alloys A and B are being used to manufacture an air craft frame. An experiment needs to be designed to compare the two in terms of maximumload capacity in tons (the maximum weight that can be tolerated without breaking).It is known that the two standard deviations in load capacity are equal at 5 tons each.An experiment is conducted in which 30 specimens of each alloy (A and B) aretested and the results recorded as follows: Xa = 49.5, Xb = 45.5, Xa – Xb = 4.The manufacturers of alloy A are convinced that this evidence shows conclusively that ?A > ?B and strongly supports the claim that their alloy is superior. Manufacturers of alloy B claim that the experiment could easily have Xa – Xb = 4 even if the two population means are equal. In other words, âthe results are inconclusive!â(a) Make an argument that manufacturers of alloy B are wrong. Do it by computingP(Xa – Xb > 4) given ?A = ?B.(b) Do you think these data strongly support alloy A?Prob 5.The strength of a plastic pipe follows normal distribution with a population variance of 6 ksi. A random sample of 25 is collected. What is the probability that a sample variance S^2(a) greater than 9.1(b) between 3.462 and 10.745.