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 ox (b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) = ox= 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. OX is more likely to be within 1 GPa of the mean in part (b). This is due to the increased variability of X that comes with a larger sample size. OX is more likely to be within 1 GPa of the mean in part (a). This is due to the increased variability of X that comes with a smaller sample size. OX is more likely to be within 1 GPa of the mean in part (b). This is due to the decreased variability of X that comes with a larger sample size. OX is more likely to be within 1 GPa of the mean in part (a). This is due to the decreased
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 ox (b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) = ox= 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. OX is more likely to be within 1 GPa of the mean in part (b). This is due to the increased variability of X that comes with a larger sample size. OX is more likely to be within 1 GPa of the mean in part (a). This is due to the increased variability of X that comes with a smaller sample size. OX is more likely to be within 1 GPa of the mean in part (b). This is due to the decreased variability of X that comes with a larger sample size. OX is more likely to be within 1 GPa of the mean in part (a). This is due to the decreased
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter13: Probability And Calculus
Section13.3: Special Probability Density Functions
Problem 30E
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Transcribed Image Text: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)
ox
=
GPa
GPa
(b) Answer the questions posed in part (a) for a sample size of n = 256 sheets.
E(X)
ox
=
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.
OX is more likely to be within 1 GPa of the mean in part (b). This is due to the increased
variability of X that comes with a larger sample size.
O X is more likely to be within 1 GPa of the mean in part (a). This is due to the increased
variability of X that comes with a smaller sample size.
OX is more likely to be within 1 GPa of the mean in part (b). This is due to the decreased
variability of X that comes with a larger sample size.
OX is more likely to be within 1 GPa of the mean in part (a). This is due to the decreased
variability of X that comes with a smaller sample size.
Transcribed Image Text:Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard
deviation of 1.3. (Round your answers to four decimal places.)
USE SALT
(a) If the distribution is normal, what is the probability that the sample mean hardness for a
random sample of 12 pins is at least 51?
(b) What is the (approximate) probability that the sample mean hardness for a random sample
of 39 pins is at least 51?
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