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According to the journal Chemical Engineering, an important property of a fiber is its water absorbency. A random sample of 20 pieces of cotton fiber was taken and the absorbency on each piece was measured. The following are the absorbency values: 18.71 21.41 20.72 21.81 19.29 22.43 20.17 23.71 19.44 20.50 18.92 20.33 23.00 22.85 19.25 21.77 22.11 19.77 18.04 21.12 (a) Calculate the sample mean and median for the above sample values. (b) Compute the 10% trimmed mean. (c) Do a dot plot of the absorbency data. (d) Using only the values of the mean, median, and trimmed mean, do you have evidence of outliers in the data? Find a formula for the probability distribution of the random variable X representing the outcome when a single die is rolled once.For Exercise 1.6 on page 13, compute the sample standard deviation in tensile strength for the samples separately for the two temperatures. Does it appear as if an increase in temperature influences the variability in tensile strength? Explain. Reference Exercise 1.6: The tensile strength of silicone rubber is thought to be a function of curing temperature. A study was carried out in which samples of 12 specimens of the rubber were prepared using curing temperatures of 20 C and 45 C. The data below show the tensile strength values in megapascals. 20"C: 2.07 2.14 2.22 2.03 2.21 2.013 2.05 2.18 2.09 2.14 2.11 2.02 45"C: 2.52 2.15 2.49 2.03 2.37 2.05 2.42 2.08 2.42 2.20 2.01 (a) Show a dot plot of the data with both low and high temperature tensile strength values. (b) Compute sample mean tensile strength for both samples. (c) Does it appear as if curing temperature has an influence on tensile strength, based on the plot? Comment further. (d) Does anything else appear to be influenced by an increase in curing temperature? Explain.\fSuppose a certain type of small data processing firm is so specialized that some have difficulty making a profit in their first year of operation. The probability density function that characterizes the proportion Y that make a profit is given by f(y) ky'(1 - y). OSys1, 10. elsewhere. (a) What is the value of k that renders the above a valid density function? (b) Find the probability that at most 50% of the firms make a profit in the first year. (c) Find the probability that at least 80% of the firms make a profit in the first year.Referring to Exercise 3.38, find (a) the marginal distribution of X; (b) the marginal distribution of Y. Reference: Exercise 3.38: If the joint probability distribution of X and Yis given by Ax, V)= an for x = 0, 1, 2, 3; y = 0, 1, 2, find a) P(X $ 2, Y= 1); b) P(X >2, Y= 1); c) (c) P(X >Y):(d) P(X + Y = 4). A coin is flipped until 3 heads in succession occur. List only those elements of the sample space that require 6 or less tosses. Is this a discrete sample space? Explain.An investment firm offers its customers municipal bonds that mature after varying numbers of years. Given that the cumulative distribution function of 7, the number of years to maturity for a randomly selected bond, is 1 3); (c) P(1.4