Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
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Chapter 4.6, Problem 89E
To determine
Whether it is comfortable to estimate population
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Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. J. of Advanced Manuf. Tech., 2010: 117–134)).
(a) If X is the sample mean Young's modulus for a random sample of
n = 16
sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution?
E(X)
=
GPa
? X
=
GPa
(b) Answer the questions posed in part (a) for a sample size of
n = 64
sheets.
E(X)
=
GPa
? X
=
GPa
(c) For which of the two random samples, the one of part (a) or the one of part (b), is X more likely to be within 1 GPa of 70 GPa? Explain your reasoning.
X is more likely to be within 1 GPa of the mean in part (a). This is due to…
Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. J. of Advanced Manuf. Tech., 2010: 117–134)).
(a) If X is the sample mean Young's modulus for a random sample of
n = 16
sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution?
E(X)
=
GPa
? X
=
GPa
(b) Answer the questions posed in part (a) for a sample size of
n = 64
sheets.
E(X)
=
GPa
? X
=
GPa
Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. J. of Advanced Manuf. Tech., 2010: 117–134)).
(a) If X is the sample mean Young's modulus for a random sample of
n = 16
sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution?
E(X)
=
GPa
? X
=
GPa
(b) Answer the questions posed in part (a) for a sample size of
n = 256
sheets.
E(X)
=
GPa
? X
=
GPa
(c) For which of the two random samples, the one of part (a) or the one of part (b), is X more likely to be within 1 GPa of 70 GPa? Explain your reasoning.
X is more likely to be within 1 GPa of the mean in part (a). This is due…
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Probability and Statistics for Engineering and the Sciences
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- Young's modulus is a quantitative measure of stiffness of an elastic material. Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. J. of Advanced Manuf. Tech., 2010: 117-134)). (a) If X is the sample mean Young's modulus for a random sample of n = 64 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) GPa GPa (b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) = GPa GPa (c) For which of the two random samples, the one of part (a) or the one of part (b), is X more likely to be within 1 GPa of 70 GPa? Explain your reasoning. X is more likely to be within 1 GPa of the mean in part (b). 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Suppose that for aluminum alloy sheets of a particular type, its mean value and standard deviation are 70 GPa and 1.6 GPa, respectively (values given in the article "Influence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis" (Intl. J. of Advanced Manuf. Tech., 2010: 117-134)). (a) If X is the sample mean Young's modulus for a random sample of n = 16 sheets, where is the sampling distribution of X centered, and what is the standard deviation of the X distribution? E(X) = GPa GPa x = (b) Answer the questions posed in part (a) for a sample size of n = 64 sheets. E(X) GPa GPa ox = (c) For which of the two random samples, the one of part (a) or the one of part (b), is X more likely to be within 1 GPa of 70 GPa? Explain your reasoning. O X is more likely to be within 1 GPa of the mean in part (b). 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This is due to the increased variability of X that comes with a larger…arrow_forwardConsider the one-way analysis of variance model Xij = µ + a; + Eij, i= 1,.., m, j= 1,..., ni, where ɛij ~ N(0,0²) are independent. Let n= n1 + · ·+ nm, ... ni 1 ni 1 X;. >Xii for i = 1,..., m, and X. = - ΣΣΧ. ni j=1 i=1 j=1 (a) Show that SS(TO) = SS(T) + SS(E), where SS(TO) = E E i=12j=1 (Xij – X..)², m m ni SS(T) Σι (X.-Χ.) and SS(E) -ΣΣ (Χ- X.) . i=1 i=1 j=1arrow_forwardSuppose data are positive numbers a1, 2, . .., T, and suppose it is decided to model these by i.i.d. random variables X1, X2, ..., X, with common Gamma(v, A) distribution as described in Definition 3.12 of the notes. Here the shape parameter v is supposed to be known, but the rate parameterdE (0, 00) is unknown and needs to be estimated. Let Å be the MLE for A. Setting i = , e;, which of the following statements is true? O i=7/v. O If Z is defined by substituting X; for æ; in A, then Z1 has a Gamma(nv, A) distribution. 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