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mathematics
advanced engineering mathematics

Questions and Answers of Advanced Engineering Mathematics

Test for exactness. If exact, solve. If not, use an integrating factor as given or obtained by inspection or by the theorems in the text. Also, if an initial condition is given, find the
Let the temperature T in a body be independent of z so that it is given by a scalar function T = T(x, t). Identify the isotherms T(x, y) = const. Sketch some of them.Sketch or graph isotherms of the
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.(-2, 3.5), (1, 2.6), (3, 1.3), (5, 0.4)
Find an augmenting path: (1) (2 (3 (6.
Does a process of producing plastic rods of length µ̃= 2 meters need adjustment if in a sample, 2 rods have the exact length and 15 are shorter and 3 longer than 2 meters? (Use the sign test.)
The OC curve in acceptance sampling cannot have a strictly vertical portion. Why?
Three specimens of high-quality concrete had compressive strength 357, 359, 413 [kg/cm2], and for three specimens of ordinary concrete the values were 346, 358, 302. Test for equality of the
Assuming normality, find a 95% confidence interval for the variance from the sample 145.3, 145.1, 145.4, 146.2.
Determine a 95% confidence interval for the mean μ of a normal population with variance σ2 = 25, using a sample of size 500 with mean 22.
Find the mean, variance, and standard derivation of the sample 21.0 21.6 19.9 19.6 15.6 20.6 22.1 22.2.
Find a 95% confidence interval for the regression coefficient Îº1, assuming (A2) and (A3) hold and using the sample.In Prob. 2Data from Prob. 4Find and graph the sample regression line of
Find a 95% confidence interval for the regression coefficient κ1, assuming (A2) and (A3) hold and using the sample.In Prob. 2Data from Prob. 2Find and graph the sample regression line of y on x and
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.x = Speed [mph]. y = Stopping distance [ft]. Also ind y at 35 mph. 30
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.x = Force [lb], y = Extension [in] of a spring. Also find the spring
Extend Prob. 5 as follows. Suppose that m times n trials were made and in the first n trials A happened k1 times, in the second n trials A happened k2 times, · · ·, in the mth n trials A
Graph the AOQ curve in Prob. 5. Determine the AOQL, assuming that rectification is applied.Data from Prob. 5Lots of copper pipes are inspected according to a sample plan that uses sample size 25 and
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.x = Sliding distance [km], y = Wear volume [mm3] 1.1 3.2 3.4 4.5 5.6
Find and graph the sample regression line of y on x and the given data as points on the same axes. Show the details of your work.x = Deformation of a certain steel [mm], y = Brinell hardness [kg/mm2]
Test for equality of population means against the alternative that the means are different assuming normality, choosing α = 5% and using two samples of sizes 12 and 18, with mean 10 and 14,
If X1 and X2 are independent normal random variables with mean 14 and 8 and variance 2 and 5, respectively, what distribution does 3 X1 - X2 have?
Find a 95% confidence interval for the variance of a normal population from the sample:Ultimate tensile strength (k psi) of alloy steel (Maraging H) at room temperature: 251, 255, 258, 253, 253, 252,
Find a 95% confidence interval for the variance of a normal population from the sample:Carbon monoxide emission (grams per mile) of a certain type of passenger car (cruising at 55 mph): 17.3, 17.8,
Do the same task as in Prob. 13 if the given sample is 0.4, 0.7, 0.2, 1.1, 0.1.Data from Prob. 13From the sample 1.9, 0.4, 0.7, 0.6, 1.4. Graph the sample distribution function F̂(x) and the
Does the random variable X = Number of accidents per week have a Poisson distribution if, within 50 weeks, 33 were accident-free, 1 accident occurred in 11 of the 50 weeks, 2 in 6 of the weeks, and
A lot of batteries for wrist watches is accepted if and only if a sample of 20 contains at most 1 defective. Graph the OC and AOQ curves. Find AOQL.
What LCL and UCL should we use instead of (1) if, instead of xÌ…, we use the sum x1+ · · · + xnof the sample values? Determine these limits in the case of Fig. 537.
If a standard medication cures about 75% of patients with a certain disease and a new medication cured 310 of the first 400 patients on whom it was tried, can we conclude that the new medication is
Test the hypothesis that, for a certain type of voltmeter, readings are independent of temperature T [°C] against the alternative that they tend to increase with T. Use a sample of values
In Prob. 11, find the mean μ, substitute it in f(x), find the maximum likelihood estimate of μ, and show that it is identical with the estimate for μ which can be obtained from that for θ in
If in a sampling plan for large lots of spark plugs, the sample size is 100 and we want the AQL to be 5% and the producer’s risk 2% what acceptance number c should we choose? (Use the normal
If a sample of 25 tires of a certain kind has a mean life of 37,000 miles and a standard deviation of 5000 miles, can the manufacturer claim that the true mean life of such tires is greater than
Find a 99% confidence interval for the mean of a normal population from the sample:Melting point (°C) of aluminum 660, 667, 654, 663, 662
Test whether a thermostatic switch is properly set to 50°C against the alternative that its setting is too low. Use a sample of 9 values, 8 of which are less than 50°C and 1 is greater.
77 students were asked to choose 3 of the integers 11, 12 ,13, · · ·, 30 completely arbitrarily. The amazing result was as follows.If the selection were completely random, the
In rolling a die, suppose that we get the first “Six” in the 7th trial and in doing it again we get it in the 6th trial. Estimate the probability p of getting a “Six” in rolling that die once.
Fifteen samples of size 100 were taken from a production of containers. The numbers of defectives (leaking containers) in those samples (in the order observed) were1 4 5 4 9 7 0 5 6 13 0 2 1 12 8From
In a clinical experiment, each of 10 patients were given two different sedatives A and B. The following table shows the effect (increase of sleeping time, measured in hours). Using the sign test,
A manufacturer claims that in a process of producing drill bits, only 2.5% of the bits are dull. Test the claim against the alternative that more than 2.5% of the bits are dull, using a sample of 400
What happens in Example 1 in the text if we increase the sample size to n = 3, leaving the other data as before? Compute P(A; 0.1) and P(A; 0.2) and compare with Example 1.
Graph λn = σ/E(R*) as a function of n. Why is λn a monotone decreasing function of n?
Determine the power of the test in Prob. 6.Data from Prob 6Assuming normality and known variance σ2 = 9, test the hypothesis μ = 60.0 against the alternative μ = 57.0 using a sample of size 20
What sample size is needed to obtain a 99% confidence interval of length 2.0 for the mean of a normal population with variance 25? Use Fig. 526. Check by calculation. 0.6 -y = 99% 0.4 Y = 95% 0.2 500
Thirty new employees were grouped into 15 pairs of similar intelligence and experience and were then instructed in data processing by an old method (A) applied to one (randomly selected) person of
Solve Prob. 5 if rolling a die 180 times gives 33, 27, 29, 35, 25, 31.Data from Prob. 5Can you claim, on a 5% level, that a die is fair if 60 trials give 1,· · ·, 6 with absolute frequencies 10,
Graph the means of the following 10 samples (thickness of gaskets, coded values) on a control chart for means, assuming that the population is normal with mean 5 and standard deviation 1.16. Time
Assuming normality and known variance σ2 = 9, test the hypothesis μ = 60.0 against the alternative μ = 57.0 using a sample of size 20 with mean x̅ = 58.50 and choosing α = 5%.
Determine a 95% confidence interval for the mean μ of a normal population with variance σ2 = 16, using a sample of size 200 with mean 74.81.
Does a process of producing stainless steel pipes of length 20 ft for nuclear reactors need adjustment if, in a sample, 4 pipes have the exact length and 15 are shorter and 3 longer than 20 ft? Use
If in 10 flips of a coin we get the same ratio as in Prob. 3 (4 heads and 6 tails), is the conclusion the same as in Prob. 3? First conjecture, then compute.Data from Prob. 3If 100 flips of a coin
Show that, in the case of the parameters α and b of the uniform distribution, the maximum likelihood estimate cannot be obtained by equating the first derivative to zero. How can we obtain maximum
What are the producer€™s and consumer€™s risks in Prob. 1 if the AQL is 2% and the RQL is 5%Data from Prob. 1Lots of kitchen knives are inspected by a sampling plan that uses a
What effect on UCL - LCL does it have if we double the sample size? If we switch from α = 1% to α = 5%
Find a 95% confidence interval for the mean of a normal population with standard deviation 4.00 from the sample 39, 51, 49, 43, 57, 59. Does that interval get longer or shorter if we take γ = 0.99
Test µ̃= 0 against µ̃> 0, using 1, -1, 1, 3, -8, 6, 0 (deviations of the azimuth [multiples of 0.01 radian] in some revolution of a satellite).
If it is known that 25% of certain steel rods produced by a standard process will break when subjected to a load of 5000 lb, can we claim that a new, less expensive process yields the same breakage
Find the maximum likelihood estimate for the parameter μ of a normal distribution with known variance σ2 = σ20 = 16
What happens in Prob. 1 if the sample size is increased to 50? First guess. Then calculate. Graph the OC curve and compare.Data from Prob. 1Lots of kitchen knives are inspected by a sampling plan
Find the probability function of X = Number of times a fair die is rolled until the first Six appears and show that it satisfies (6).
If the life of ball bearings has the density f(x) = ke-x if 0 ≤ x ≤ 2 and 0 otherwise, what is k? What is the probability P(X ≥ 1)?
Prove that x̅ must always lie between the smallest and the largest data values.
Of a lot of 12 items, 3 are defective.(a) Find the number of different samples of 3 items. Find the number of samples of 3 items containing(b) No defectives(c) 1 defective(d) 2 defectives(e) 3
Show that(a) Show that E(X - μ) = 0, σ2 = E(X2) - μ2.(b) Prove (10)-(12).(c) Find all the moments of the uniform distribution on an interval α ≤ x ≤ b.(d) The skewness ϒ of a random variable
Prove the statement involving (18).
Using a Venn diagram, show that A ⊆ B if and only if A ∩ B = A
Do the same tasks as in Prob. 17 for the hardness data in Prob. 8.Data from Prob. 174 1 3 10 2.
Using a Venn diagram, show that A ⊆ B if and only if A ∩ B = A.
Let (X, Y) have the densityf(x, y) = k if x2 + y2 < 1and 0 otherwise. Determine k. Find the densities of the marginal distributions. Find the probabilityP(X2 + Y2 < 1/4).
Extending Theorem 4, show that P(A ∩ B ∩ C) = P(A)P(B|A)P(C|A ∩ B).
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the Brinell hardness data in Prob. 8Data from Prob. 886 86
The moment generating function G(t) is defined byorwhere X is a discrete or continuous random variable, respectively.(a) Assuming that term wise differentiation and differentiation under the integral
What are the outcomes in the sample space of the experiment of simultaneously tossing three coins?
What capacity must the tank in Prob. 15 have in order that the probability that the tank will be emptied in a given week be 5%?Data from Prob. 15A small filling station is supplied with gasoline
Prove (2).
(a) Find the points of inflection of the curve of (1).(b) Considering Î¦2(ˆž) and introducing polar coordinates in the double integral (a standard trick worth remembering),
In drawing 2 screws from a lot of right-handed and left-handed screws, let A, B, C, D mean at a least 1 right-handed, at least 1 left-handed, 2 right-handed, 2 left-handed, respectively. Are A and B
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the iron contents in Prob. 4Data from Prob. 472.8 70.4
If a ticket office can serve at most 4 customers per minute and the average number of customers is 120 per hour, what is the probability that during a given minute customers will have to wait?
Find the mean, standard deviation, and variance in Prob. 12.Data from Porb 1213.5, 13.2, 12.1, 13.6, 13.3.
An electronic device consists of two components. Let X and Y [years] be the times to failure of the first and second components, respectively. Assume that (X, Y) has the density f(x, y) = 4e-2(x+y)
(a) Prove Theorem 2.(b) Prove the last statement of Theorem 3.(c) Derive (11) from (8).(d) By the binomial theorem,
Suppose that in a production of spark plugs the fraction of defective plugs has been constant at over a long time and that this process is controlled every half hour by drawing and inspecting two
Graph a sample space for the experiments:List all 8 subsets of the sample space S = {α, b, c}.
Find the mean and compare it with the median. Find the standard deviation and compare it with the interquartile range.For the phone call data in Prob. 2Data from Prob. 26 6
Find P(X > 4, Y > 4) and P(X ≤ 1, Y ≤ 1) if (X, Y) has the density f(x, y) = 1/32 if x ≥ 0, y ≥ 0, x + y ≤ 8.
Suppose that a certain type of magnetic tape contains, on the average, 2 defects per 100 meters. What is the probability that a roll of tape 300 meters long will contain(a) x defects(b) No defects?
Same task as in Prob. 11. for the data 13.5, 13.2, 12.1, 13.6, 13.3.Data from Prob. 11Make a stem-and-leaf plot, histogram, and boxplot of the data 110, 113, 109, 118, 110, 115, 104, 111, 116, 113.
What total sum can you expect in rolling a fair die 20 times? Do the experiment. Repeat it a number of times and record how the sum varies.
If the mean weight of certain (empty) containers is 5 lb the standard deviation is 0.2 lb, and if the filling of the containers has mean weight 100 lb and standard deviation 0.5 lb, what are the mean
If 3 suspects who committed a burglary and 6 innocent persons are lined up, what is the probability that a witness who is not sure and has to pick three persons will pick the three suspects by
If a circuit contains four automatic switches and we want that, with a probability of 99%, during a given time interval the switches to be all working, what probability of failure per time interval
A producer sells electric bulbs in cartons of 1000 bulbs. Using (11), find the probability that any given carton contains not more than 1% defective bulbs, assuming the production process to be a
If the diameter X of axles has the density f(x) = k if 119.9 ≤ x ≤ 120.1 and 0 otherwise, how many defectives will a lot of 500 axles approximately contain if defectives are axles slimmer than
Let p = 2% be the probability that a certain type of light bulb will fail in a 24-hour test. Find the probability that a sign consisting of 15 such bulbs will burn 24 hours with no bulb failures.
Represent the data by a stem-and-leaf plot, a histogram, and a boxplot:-0.51 0.12 -0.47 0.95 0.25 -0.18 -0.54
If, in Prob. 9, a defective bolt is one that deviates from 1.00 cm by more than 0.06 cm, what percentage of defectives should we expect?Data from Prob. 9If the diameter X [cm] of certain bolts has
Using Theorem 1, obtain the formula for the mean of the hypergeometric distribution. Can you use Theorem 3 to obtain the variance of that distribution?
If a cage contains 100 mice, 3 of which are male, what is the probability that the 3 male mice will be included if 10 mice are randomly selected?
The breaking strength X [kg] of a certain type of plastic block is normally distributed with a mean of 1500 kg and a standard deviation of 50 kg. What is the maximum load such that we can expect no
Graph a sample space for the experiments:In rolling 3 dice, are the events A: Sum divisible by 3 and B: Sum divisible by 5 mutually exclusive?
Graph a sample space for the experiments:Choosing a committee of 2 from a group of 5 people

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