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introduction to statistical investigations

Questions and Answers of Introduction To Statistical Investigations

6.5 Generate samples of a linear regression model with different distribution of errors and apply the function LADLS to these data.
6.4 The -trimmed mean is asymptotically efficient for f in the Bahadur sense if and only if the central part of distribution F, namely 100(1 −2)%, is normal, while the distribution has 100% of
1.1 When two objects of material are attached together with glue, the glue can be oxidized in the moment just before the attachment to increase adhesion. It is done by blowing gas on to the glue.
1.2 Consider Example 1.1 where a bar is attached to a wall. Denote the end point position (yx, yy), the length of the bar L, the attachment point in the wall Z, and the position of the support
2.1 Consider the strip position problem. In Section 2.3 the fact that there actually are 10 replicates was used, but not in Section 2.2. Use the entire set of data given in Table B.2 to check out
2.2 For the strip position data of Example 2.1, the temperature is the noise factor.The problem was analysed in order to find out how to design in order to get a strip position that is robust against
2.3 Once an experiment is performed, it is a good habit to look at the data graphically to see if there are any data points that look strange. Try to find a way to plot the data of the strip
2.4 The change in temperature between a heating element and the next operation in an industrial process depends on a range of factors, such as the distance between the heating element and the
2.5 Consider the interaction plot of Figure 2.12. How can the information that the plot provides be used to obtain robustness if:(a) The control factor is numerical and continuous, as, for example, a
3.1 What could the noise factors be in these cases?(a) You are about to design a screw cap to a bottle. You want it to be easy to open and have chosen to look at the opening energy as the response.
4.1 Find ideal functions for some of these:(a) An electrical motor.(b) A hose clamp.(c) A moulding process.(d) A strain gauge sensor.(e) Something from your own field of work.
4.2 A team is about to design a sterilization mechanism where a chemical in gas form is used to inactivate bugs. They are not certain how to set up the P diagram and need some help. The factors,
4.3 A team has decided to make a robust design study on a simple cap of the type used for PET bottles. The team wants to avoid a situation where the consumers find it unpleasant to open, so
4.4 Recall the rechargeable battery of Example 4.3. The fact that the output voltage may be too small to be useful was left unconsidered. This issue can be handled in different ways.(a) Consider the
4.5 Consider the problem with the wheel to transport boxes in Example 3.6. Since gravity is used to make the boxes move, there is actually one energy transformation involved.(a) Express the problem
5.3 Consider a problem with four control factors and three noise factors. Each factor has two levels.(a) In crossed arrays, howmany runs are needed to avoid two-factor interactions to be confounded
5.4 Set up an inner and outer array with four factors on two levels each, three control factors A, B, and C, and one noise factor D. Suppose that the true model iswhere ???? = 1. Generate random data
6.3 White spots. A paper, a postcard, or a poster that is printed is never perfectly flat. There are cavities. When printed, the ink cannot cover the cavities. It causes white spots, a failure mode
7.2 In a production process, some process parameters are difficult to control exactly.In parts of the chemical industry it can typically be pressure and temperature values. The nominal value can be
8.3 (Example 8.5). Find the Pareto optimal set ofwith respect to type IV optimal robustness, subject tounder the assumption that E[z] = 0. y = 10+ 1.7z+1.7x12 -0.3x2z, y2 3+0.8z 1.3x1z + 0.9x2z, -
9.1 Assume that an experimentalist has a list of seven possible factors and is about to run a screening design to identify the most vital ones for further experimentation.Write a code in some
9.2 Recall Example 9.1. Use cross validation by excluding one observation at a time from the data and estimate the optimal value of the window width ???? by minimizing PRESS, the prediction error sum
9.3 Since robust design is about preventing variation to propagate, we want to find values of the control variables x so that (g′z)2 inis small. The quantity (g′z)2 takes its minimum whenApply
A.2 Lorenzen and Anderson (1993). The intermediate shaft steering column (Figure A.4) connects the steering wheel to the power steering motor. In the manufacture, the tube, which has a diameter
A.1 Consider the steering wheel torque in Example A.1.(a) Calculate the average loss without using the estimated model. Compare with the results of Example A.1. If possible, divide this average loss
11.1 Taguchi’s L18(21 × 37) array is given by Table 11.2.(a) Under the assumption that the factors are numerical, which linear combination of main effects will the interaction between factors A
8.4 Use Taylor expansions of g(x, z) around z = E[Z] to discuss Equation (8.7).Apply this to Equation (8.2) and comment upon the conclusions.
8.2 Consider Example 8.8. Suppose that the target value is g = 10. What values should the control variables x1 and x2 have in order to both be robust against the noise z and aim at the target?
8.1 Consider Example 8.2. Find the value x that gives type III optimal robustness.
7.3 (a) Consider Example 7.4. Keeping in mind the fact that the slope should be maximized, could this type of problem be considered as larger-the-better?(b) Analyse the data expressing the variation
7.1 Build a model from the data of Table 7.1 and compare it to Equation (7.1).
6.4 Consider the wave soldering problem in Example 6.4.(a) Analyse the data by separately minimizing log ????y, log ????v, and log ????y and maximizing log ????v. Are there any contradictory
6.2 Consider the stress data of Example 6.2.(a) Since there are two replicates, it is possible to estimate the effects each factor has on log ????. Perform this analysis.(b) How can the product be
6.1 Consider the data of Example 6.1.(a) Identify the significant effects and build a statistical model.(b) Analyse the data in terms of the logarithm of the loss, log L. Will the conclusions be the
5.2 A milk package is usually, formed by making a sleeve tube of a carton sheet, sealing it along the tube (the longitudinal sealing) and in one end, and then after filling also in the other end, the
5.1 Analyse the wafer data of Table C.1 not just graphically but also analytically.Use the analytical method you are used to and prefer (e.g. the PRESS value, the predictive R2, or the p values from
3.6 In Example 3.9, the robustness to age for some property of a nozzle was discussed.How many units (nozzles) are needed for the test?
3.5 One way to set up the experiment with the injection moulding of Example 3.11 could be the one shown in Table 3.7. Analyse the data.
3.4 Analyse the data of Table 3.5.
3.3 Recall Example 3.8, the slip-off time of the boxes. As mentioned, the noise factor might be difficult to control even in the test environment and the method described there may be too inaccurate.
3.2 In Exercise 3.1, noise factors were identified. For one of the problems there:(a) Use your engineering knowledge and experience to judge whether anything can be compounded.(b) Use your knowledge
Consider an arbitrary probability measure space \((\Omega, \mathcal{F}, P)\) and let \(X_{r}\) be the collection of all possible random variables \(X\) that map \(\Omega\) to \(\mathbb{R}\) subject
Within the context of Exercise 1, let \(\|X\|_{r}\) be defined for \(X \in X_{r}\) as\[\|X\|_{r}=\left[\int_{\Omega}|X(\omega)|^{r} d P(\omega)ight]^{1 / r}\]Prove that \(\|X\|_{r}\) is a norm. That
Let \(X_{1}, \ldots, X_{n}\) be a set of independent and identically distributed random variables from a distribution \(F\) that has parameter \(\theta\). Let \(\hat{\theta}_{n}\) be an unbiased
Consider a sequence of random variables \(\left\{X_{n}ight\}_{n=1}^{\infty}\) where \(X_{n}\) has probability distribution function\[f_{n}(x)= \begin{cases}{[\log (n+1)]^{-1}} & x=n \\ 1-[\log
Suppose that \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is a sequence of independent random variables from a common distribution that has mean \(\mu\) and variance \(\sigma^{2}\), such that
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables that converge in \(r^{\text {th }}\) mean to a random variable \(X\) as \(n ightarrow \infty\) for some \(r>0\). Prove that
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has probability distribution function\[f_{n}(x)= \begin{cases}1-n^{-\alpha} & x=0 \\
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(P\left(\left|X_{n}ight| \leq Yight)=1\) for all \(n \in \mathbb{N}\) where \(Y\) is a positive integrable
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that\[E\left(\sup _{n \in \mathbb{N}}\left|X_{n}ight|ight)
Prove that if \(a, x\), and \(y\) are positive real numbers then\[2 \max \{x, y\} \delta\{2 \max \{x, y\} ;(a, \infty)\} \leq 2 x \delta\left\{x ;\left(\frac{1}{2}a, \inftyight)ight\}+2 y
Suppose that \(\left\{X_{n}ight\}_{n=1}^{\infty}\) is a sequence of random variables such that \(X_{n} \xrightarrow{r} X\) as \(n ightarrow \infty\) for some random variable \(X\). Prove that for
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has an \(\operatorname{ExpONEntial}\left(\theta_{n}ight)\) distribution for all \(n \in
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has a Triangular \(\left(\alpha_{n}, \beta_{n}, \gamma_{n}ight)\) distribution for all \(n \in
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has a \(\operatorname{BETA}\left(\alpha_{n}, \beta_{n}ight)\) distribution for all \(n \in
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables where \(X_{n}\) has distribution \(F_{n}\) which has mean \(\theta_{n}\) for all \(n \in \mathbb{N}\). Suppose that\[\lim
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(X_{n}\) has distribution \(F_{n}\) for all \(n \in \mathbb{N}\). Suppose that \(X_{n} \xrightarrow{\text { a.c.
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(X_{n}\) has a \(\mathrm{N}\left(0, \sigma_{n}^{2}ight)\) distribution, conditional on \(\sigma_{n}\). For each
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(X_{n}\) has a \(\mathrm{N}\left(\mu_{n}, \sigma_{n}^{2}ight)\) distribution, conditional on \(\mu_{n}\) and
Write a program in \(\mathrm{R}\) that simulates a sequence of independent and identically distributed random variables \(X_{1}, \ldots, X_{100}\) where \(X_{n}\) follows a distribution \(F\) that is
Write a program in \(\mathrm{R}\) that simulates a sequence of independent and identically distributed random variables \(B_{1}, \ldots, B_{500}\) where \(B_{n}\) is a
Write a program in \(\mathrm{R}\) that simulates a sequence of independent random variables \(X_{1}, \ldots, X_{100}\) where \(X_{n}\) as probability distribution function\[f_{n}(x)=
Write a program in \(\mathrm{R}\) that simulates a sequence of independent random variables \(X_{1}, \ldots, X_{100}\) where \(X_{n}\) is a \(\mathrm{N}\left(0, \sigma_{n}^{2}ight)\) random variable
Write a program in \(\mathrm{R}\) that simulates a sequence of independent random variables \(X_{1}, \ldots, X_{100}\) where \(X_{n}\) is a \(\mathrm{N}\left(\mu_{n}, \sigma_{n}^{2}ight)\) random
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has a \(\operatorname{GAmma}\left(\theta_{n}, 2ight)\) distribution where
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent random variables where \(X_{n}\) has a \(\operatorname{BERNOULLi}\left(\theta_{n}ight)\) distribution where
In the context of Theorem 6.1 (Lindeberg, Lévy, and Feller), prove that Equation (6.3) implies Equation (6.2). Theorem 6.1 (Lindeberg, Lvy, and Feller). Let {X} be a sequence of independent random
Prove Corollary 6.2. That is, let \(\left\{\left\{X_{n k}ight\}_{k=1}^{n}ight\}_{n=1}^{\infty}\) be a triangular array where \(X_{11}, \ldots, X_{n 1}\) are mutually independent random variables for
Let \(\left\{\left\{X_{n, k}ight\}_{k=1}^{n}ight\}_{n=1}^{\infty}\) be a triangular array of random variables where \(X_{n, k}\) has a \(\operatorname{BERnOulli}\left(\theta_{n, k}ight)\)
Prove Theorem 6.4. That is, suppose that \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(\sigma_{n}^{-1}\left(X_{n}-\muight) \xrightarrow{d} Z\) as \(n ightarrow
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(\sigma_{n}^{-1}\left(X_{n}-\muight) \xrightarrow{d} Z\) as \(n ightarrow \infty\) where \(Z\) is a
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of random variables such that \(\sigma_{n}^{-1}\left(X_{n}-\muight) \xrightarrow{d} Z\) as \(n ightarrow \infty\) where \(Z\) is a
Let \(\left\{X_{n}ight\}_{n=1}\) be a set of independent and identically distributed random variables from a distribution with me \(\mu\) and finite variance \(\sigma^{2}\). Show that\[S^{2}=n^{-1}
In Example 6.6, find \(\boldsymbol{\Lambda}\) and \(\mathbf{d}^{\prime}(\theta) \boldsymbol{\Lambda} \mathbf{d}(\theta)\). Example 6.6. Let {X} be a sequence of independent and identically
Let \(\left\{\mathbf{X}_{n}ight\}\) be a sequence of \(d\)-dimensional random vectors where \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{Z}\) as \(n ightarrow \infty\) where \(\mathbf{Z}\) has a
Let \(\left\{\mathbf{X}_{n}ight\}\) be a sequence of \(d\)-dimensional random vectors where \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{Z}\) as \(n ightarrow \infty\) where \(\mathbf{Z}\) has a
Let \(\left\{\mathbf{X}_{n}ight\}\) be a sequence of \(d\)-dimensional random vectors where \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{Z}\) as \(n ightarrow \infty\) where \(\mathbf{Z}\) has a
Let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of two-dimensional random vectors where \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{Z}\) as \(n ightarrow \infty\) where \(\mathbf{Z}\)
Let \(\left\{\mathbf{X}_{n}ight\}_{n=1}^{\infty}\) be a sequence of three-dimensional random vectors where \(\mathbf{X}_{n} \xrightarrow{d} \mathbf{Z}\) as \(n ightarrow \infty\) where \(\mathbf{Z}\)
Write a program in \(\mathrm{R}\) that simulates 1000 samples of size \(n\) from an EXPONENTIAL(1) distribution. On each sample compute \(n^{1 / 2}\left(\bar{X}_{n}-1ight)\) and \(n^{1 /
Write a program in \(\mathrm{R}\) that simulates 1000 observations from a Multino\(\operatorname{MIAL}(n, 3, \mathbf{p})\) distribution where \(\mathbf{p}^{\prime}=\left(\frac{1}{4}, \frac{1}{4},
Write a program in \(\mathrm{R}\) that simulates 1000 sequences of independent random variables of length \(n\) where the \(k^{\text {th }}\) variable in the sequence has an
Write a program in \(\mathrm{R}\) that simulates 1000 samples of size \(n\) from a UNI\(\operatorname{FORm}\left(\theta_{1}, \theta_{2}ight)\) distribution where \(n, \theta_{1}\), and \(\theta_{2}\)
Let \(f\) be a real function and define the Fourier norm as Feller (1971) does as\[(2 \pi)^{-1} \int_{-\infty}^{\infty}|f(x)| d x\]For a fixed value of \(x\), is this function a norm?
Prove that the Fourier transformation of \(H_{k}(x) \phi(x)\) is \((i t)^{k} \exp \left(-\frac{1}{2} t^{2}ight)\). Hint: Use induction and integration by parts as in the partial proof of Theorem 7.1.
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables where \(X_{n}\) has a \(\operatorname{Gamma}(\alpha, \beta)\) distribution for all
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables where \(X_{n}\) has a \(\operatorname{BETA}(\alpha, \beta)\) distribution for all
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables where \(X_{n}\) has a density that is a mixture of two NORMAL densities of the form
Prove Theorem 7.8. That is, let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\). Let \(F_{n}(t)=P\left[n^{1 /
Let \(\left\{R_{n}ight\}_{n=1}^{\infty}\) be a sequence of real numbers such that \(R_{n}=o\left(n^{-1}ight)\) as \(n ightarrow \infty\). Prove that \(R_{n}^{2}=o\left(n^{-1}ight)\) as \(n ightarrow
Suppose that \(v_{1}(\alpha)\) and \(v_{2}(\alpha)\) are constant with respect to \(n\). Prove that if \(R_{n}=\left[n^{-1 / 2} v_{1}(\alpha)+n^{-1} v_{2}(\alpha)+o\left(n^{-1}ight)ight]^{2}\) then a
Suppose that \(g_{\alpha, n}=v_{0}(\alpha)+n^{-1 / 2} v_{1}(\alpha)+n^{-1} v_{2}(\alpha)+o\left(n^{-1}ight)\) as \(n ightarrow \infty\) where \(v_{0}(\alpha), v_{1}(\alpha)\), and \(v_{2}(\alpha)\)
Prove that\[\int_{-\infty}^{\infty} \exp (t x) \phi(x) H_{k}(x) d x=t^{k} \exp \left(\frac{1}{2} t^{2}ight)\]
Suppose that \(X_{1}, \ldots, X_{n}\) are a set of independent and identically distributed random variables from a distribution \(F\) that has mean equal to zero, unit variance, and cumulant
Suppose that \(X_{1}, \ldots, X_{n}\) are a set of independent and identically distributed random variables from a distribution \(F\) that has mean equal to\(\theta\), variance equal to
Using Equation (7.28), prove that \(-r_{1}(x)=\frac{1}{6} ho_{3} H_{2}(x)\) and \[-r_{2}(x)=\frac{1}{24} ho_{4} H_{3}(x)+\frac{1}{72} ho_{3}^{2} H_{5}(x) .\] d (x)pk(x) = (x)rk(x), dx (7.28)
Let \(\left\{X_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\). Let \(F_{n}(t)=P\left[n^{1 / 2}
Let \(\left\{W_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed random variables from a distribution \(F\) with mean \(\eta\) and variance \(\theta\). Prove that
In the context of Example 7.8, let \(\left\{W_{n}ight\}_{n=1}^{\infty}\) be a sequence of independent and identically distributed bivariate random vectors from a distribution \(F\) having mean vector
Prove that the polynomials given in Equations (7.44) and (7.46) reduce to those given in Equations (7.29) and (7.30), when \(\theta\) is taken to be the univariate mean. -11(x) = 103H2(x) (7.29)

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