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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?Accurate airflow measurement requires straight unobstructed pipe for a minimum of 10 diameters upstream and 5 diameters downstream of the mea- suring device. In a field application, physical con- straints compromise the pipe layout, so the engineer is considering installing the airflow probes in an elbow, knowing that flow measurement will be less accurate but good enough for process control. This is plan 1, which will be in place for only 3 years, after which a more accurate flow measurement sys- tem with the same costs as plan I will be available. This plan will have a first cost of $26,000 with an annual maintenance cost estimated at $5000 Plan 2 involves installation of a recently de- signed submersible airflow probe. The stainless steel probe can be installed in a drop pipe with the transmitter located in a waterproof enclosure on the handrail. The first cost of this system is $83,000, but because it is accurate and more durable, it will not have to be replaced for at least 6 years. Its main- tenance cost is estimated to be $1400 per year plus $2500 in year 3 for replacement of signal process- ing software. Neither system will have a salvage value. At an interest rate of 10% per year, which one should be selected on the basis of a present worth comparison?Consider a small but known volume of metal that has a large thermal conductivity. (a) Since the thermal conductivity is large, spatial temperature gradients that develop within the metal in response to mild heating are small. Neglecting spatial temperature gradients, derive a differential equation that could be solved for the temperature of the metal versus time 7() if the metal is sub- jected to a fixed surface heat rate q supplied by an electric heater. (b) A student proposes to identify the unknown metal by comparing measured and predicted thermal responses. Once a match is made, relevant thermo- physical properties might be determined, and, in turn, the metal may be identified by comparison to published property data. Will this approach work? Consider aluminum, gold, and silver as the candi- date metals.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.Let X denote the number of times a certain numerical control machine will malfunction: 1, 2, or 3 times on any given day. Let Y denote the number of times a technician is called on an emergency call. Their joint probability distribution is given as f(x, y) 2 3 0.05 0.05 0.10 0.05 0.10 0.35 0.00 0.20 0.10 (a) Evaluate the marginal distribution of X. (b) Evaluate the marginal distribution of Y. (c) Find P(Y = 3 [ X = 2).The following scores represent the final examination grades for an elementary statistics course: 60 32 57 74 70 54 64 89 76 48 67 17 69 74 80 85 (a) Construct a stem-and-leaf plot for the examination grades in which the stems are 1, 2, 3, 9 (b) Construct a relative frequency histogram, draw an estimate of the graph of the distribution, and discuss the skewness of the distribution. (c) Compute the sample mean, sample median, and sample standard deviation