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Compute the sample variance and standard deviation for the water absorbency data of Exercise 1.2 on page 13. Reference: Exercise 1.2 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?A pan is used to boil water by placing it on a stove, from which heat is transferred at a fixed rate q. There are two stages to the process. In Stage 1, the water is taken from its initial (room) temperature 7, to the boiling point, as heat is transferred from the pan by natural convection. During this stage, a constant value of the convection coef- ficient h may be assumed, while the bulk temperature of the water increases with time, Iz = 7_(n). In Stage 2, the water has come to a boil, and its temperature remains at a fixed value, Tz = 7, as heating continues. Consider a pan bottom of thickness L and diameter D, with a coordi- nate system corresponding to x = 0 and x = L for the sur- faces in contact with the stove and water, respectively. (a) Write the form of the heat equation and the boundary/ initial conditions that determine the variation of temperature with position and time, 7(x, /), in the pan bottom during Stage 1. Express your result in terms of the parameters q., D. L. h, and Tz, as well as appropriate properties of the pan material. b) During Stage 2. the surface of the pan in contact with the water is at a fixed temperature, 7(L, () = Tz > T. Write the form of the heat equation and boundary conditions that determine the temperature distribution 7(x) in the pan bottom. Express your result in terms of the parameters q., D. L, and Tr, as well as appropriate properties of the pan material.Stereophonics, Inc., plans to borrow $20,000 from a bank for 1 year at 9% interest for new recording equipment. (@) Compute the interest and the total amount due after 1 year. (b) Con- struct a column graph that shows the original loan amount and total amount due after 1 year used to compute the loan interest rate of 9% per year.An important factor in solid missile fuel is the particle size distribution. Significant problems occur if the particle sizes are too large. From production data in the past, it has been determined that the particle size (in micrometers) distribution is characterized by f(x) = [3r, r>1, elsewhere. (a) Verify that this is a valid density function. (b) Evaluate F(x). (c) What is the probability that a random particle from the manufactured fuel exceeds 4 micrometers? Show that the n pieces of information in _ (x - ?) are not independent; that is, show that 2(n- 1) =0The waiting time, in hours, between successive speeders spotted by a radar unit is a continuous random variable with cumulative distribution function Find the probability of waiting less than 12 minutes between successive speeders (a) using the cumulative distribution function of X; (b) using the probability density function of X. Determine the value c so that each of the following functions can serve as a probability distribution of the discrete random variable X: (a) f(x) = c(x2 + 4), for x = 0, 1, 2, 3; (b) f(x) = c() (->), for x = 0, 1, 2.The proportion of the budget for a certain type of industrial company that is allotted to environmental and pollution control is coming under scrutiny. A data collection project determines that the distribution of these proportions is given by f(y) = 5(1-V)'. OSys1. elsewhere. (a) Verify that the above is a valid density function. (b) What is the probability that a company chosen at random expends less than 10% of its budget on environmental and pollution controls? (c) What is the probability that a company selected at random spends more than 50% of its budget on environmental and pollution controls?\f