1. Traffic speed. The following is the frequency distribution of travel times of motor-cars on a highway.
Mean times (min): 53, 58. 63, 68, 73. 78. 83. 88, 93, 98, 103, 108.113. 118, 123,128, 133, 138, 143, 148, 153, 158, 163, 168.
Corresponding frequencies: 10, 24, 109, 12(p+8), 12(p+3), 11p, 97, 102, 104, 92, 68, 72, 66, 61, 36, 33, 17, 15, 10, 8, 9, 6, 7, 3.
Draw the histogram. Now many peaks arc there and what they tell us? What inference can be made from the mean time interval between the peaks?
What is the variance and standard deviation? What is mean by variation in this data?
2. A dealer in recycled paper places empty trailers behind foods locations. The trailers arc gradually filled by customers. The dealer picks up the trailers every other week. This schedule works as long as the average amount of recycled paper is more than 1600 cubic feet (the amount needed to justify operating costs of the trailer). The dealer's records for 1K 2-week periods show the following volumes:
1820, 1590, 1440, 1730, 1680, 1750, 1720, 18pq, 1900,
1570, 1700, 1900, 1800, 1770, 2010, 1580, 1620, 1690.
(a) Write the hypothesis.
(b) What distribution (test statistic) is to be used? Write reasons.
(c) Calculate the test statistic.
(d) Build the 95% CI and check the value of the statistic inside the interval if it lies?
(c) Identify the rejection region for the testing distribution.
(e) Construct critical region and draw your conclusion using alpha = 0.05.
(f) What are the implications of making the wrong decision?
3. Plot the given data on graph paper to get scatter diagram and then determine regression line of y on x. Graph this regression line on the same scatter diagram. Find the sum of distances from the line to the points. Explain what method of Least squares is.
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Density
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2411
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2415
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2425
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2427
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244p
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247(p+2)
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2480
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2481
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2483
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2487
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Compressive strength
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49.9
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50.7
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52.5
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53.2
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57.p
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58.(p+2)
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60.1
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601
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60.5
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60.9
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4. (a) Evaluate the double integral ∫D∫(px2 +qy + 1)dA, where D is the region bounded b
(i)y = x2 and x = y2 (ii) y = x, x = 1, and y = p. (iii) y = x2 and y = p - x2
(iv) px + pq and the coordinate axes in the first quadrant.
(v) Half circle x2 + y2 = 1 in the upper half plane and y = 0.
(b) Evaluate the integral in part (a) by changing the order of integration in each case.
5. Construct the vector field F→ = -pyi→ + qxj→ by finding at least 10 vectors. Find the Flux through (i) the square with side p (ii) a circle with radius p (iii) a triangle with vertices (-p, 0), (p, 0), (0, q). Find also the Curl F→ and explain what is mean by Curl F→ .
6. Evaluate c∫F→ (rv)·dr→ , F→ = y2i→ +x2j→ counterclockwise around the boundary of C of the indicated region R:
(i) 1 ≤ x2 + y2 ≤ p+1, x ≥ 0, y ≥ 0.
(ii) 1 ≤ x2 +y2 ≤ p +1, x ≥ 0, y ≤ 0.
(iii) Square with vertices (0, 0), (p, 0), (p, p), (0, p).
(iv) Triangle with vertices (-p, 0), (0, 0), (0, p)
(v) Circle with radius 2 and center (-p, p).
7. Find the volume of a tank whose base dimensions are 0 ≤ x ≤ p, 0 ≤ y ≤ q and its top is given by z = px + qy +10 using (i) double integrals (ii) triple integrals. Draw also a clear sketch of this tank. (3D Plot)
8. Find the general solution of the first order differential equation and graph it for at least 10 values of C using desmos graphing calculator or any other software for graphs. (This graph is called direction field).
dy/dx = pq - 2x + 1
Find then the particular solution using the condition y(p) = q . Draw the graph of this solution on the same graph.