Probability and Statistics for Engineering and the Sciences
9th Edition
ISBN: 9781305251809
Author: Jay L. Devore
Publisher: Cengage Learning
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Chapter 7, Problem 56SE
The accompanying data on crack initiation depth (μm) was read from a lognormal probability plot that appeared in the article “Incorporating Small Fatigue Crack Growth in Probabilistic Life Prediction: Effect of Stress Ratio in Ti-6Al-2Sn-6Mo” (Intl. J. of Fatigue, 2013: 83–95). Although the pattern in the plot was quite straight, a normal probability plot of the data also shows a reasonably linear pattern. And a boxplot indicates that the distribution is quite symmetric in the middle 50% of the data and only mildly skewed overall. It is therefore reasonable to estimate and predict using t intervals.
- a. Estimate the true average crack initiation depth with a 99% CI and interpret the resulting interval.
- b. Predict the value of a single crack initiation depth by constructing a 99% PI.
- c. Interpret in context the
meaning of 99% in (b).
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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…
Chapter 7 Solutions
Probability and Statistics for Engineering and the Sciences
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