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Chapter 5: Q46E (page 236)

Young鈥檚 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 \({\rm{70 GPa}}\) and \({\rm{1}}{\rm{.6 GPa}}\), respectively (values given in the article 鈥淚nfluence of Material Properties Variability on Springback and Thinning in Sheet Stamping Processes: A Stochastic Analysis鈥� (Intl. J. of Advanced Manuf. Tech., \({\rm{2010:117 - 134}}\))).

  1. If \({\rm{\bar X}}\) is the sample mean young鈥檚 modulus for a random sample of \({\rm{n = 16}}\)sheets, where is the sampling distribution of \({\rm{\bar X}}\)centered, and what is the standard deviation of the \({\rm{\bar X}}\)distribution?
  2. Answer the questions posed in part (a) for a sample size of \({\rm{n = 64}}\)sheets.
  3. For which of the two random samples, the one of part (a) or the one of part (b), is \({\rm{\bar X}}\) more likely to be within \({\rm{1GPa}}\) of \({\rm{70 GPa}}\)? Explain your reasoning.

Short Answer

Expert verified

a. \(\begin{array}{*{20}{c}}&{{{\rm{\mu }}_{{\rm{\bar x}}}}{\rm{ = 70GPa}}}\\&{{{\rm{\sigma }}_{{\rm{\bar x}}}}{\rm{ = 0}}{\rm{.4GPa}}}\end{array}\)

b. \(\begin{array}{*{20}{c}}&{{{\rm{\mu }}_{{\rm{\bar x}}}}{\rm{ = 70GPa}}}\\&{{{\rm{\sigma }}_{{\rm{\bar x}}}}{\rm{ = 0}}{\rm{.2GPa}}}\end{array}\)

c. Part (b) \({\rm{n = 64}}\) is more likely.

Step by step solution

01

Definition of young’s modulus

Young's Modulus (also known as Elastic Modulus or Tensile Modulus) is a mechanical property of linear elastic solids such as rods, wires, and other similar objects. Other quantities, such as Bulk modulus and shear modulus, can be used to determine a material's elastic properties, but Young's Modulus is the most widely utilised.

02

Calculating the standard deviation of the \({\rm{\bar X}}\) distribution

Given:

\(\begin{aligned}&{{\rm{\mu = 70GPa}}}\\&{{\rm{\sigma = 1}}{\rm{.6GPa}}}\\&{{\rm{n = 16}}}\end{aligned}\)

The sample mean's sampling distribution has a mean of \({\rm{\mu }}\) and a standard deviation of \(\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}\)

\(\begin{aligned}&{{{\rm{\mu }}_{{\rm{\bar x}}}}{\rm{ = \mu = 70GPa}}}\\&{{{\rm{\sigma }}_{{\rm{\bar x}}}}{\rm{ = }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}{\rm{ = }}\frac{{{\rm{1}}{\rm{.6GPa}}}}{{\sqrt {{\rm{16}}} }}{\rm{ = 0}}{\rm{.4GPa}}}\end{aligned}\)

03

Calculating the part (a) for a sample size of \({\rm{n = 64}}\) sheets. 

Given:

\({\rm{\mu = 70GPa}}\)

\({\rm{\sigma = 1}}{\rm{.6GPa}}\)

\({\rm{n = 64}}\)

The sample mean's sampling distribution has a mean of \({\rm{\mu }}\) and a standard deviation of \(\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}\)

\(\begin{aligned}&{{{\rm{\mu }}_{{\rm{\bar x}}}}{\rm{ = \mu = 70GPa}}}\\&{{{\rm{\sigma }}_{{\rm{\bar x}}}}{\rm{ = }}\frac{{\rm{\sigma }}}{{\sqrt {\rm{n}} }}{\rm{ = }}\frac{{{\rm{1}}{\rm{.6GPa}}}}{{\sqrt {{\rm{64}}} }}{\rm{ = 0}}{\rm{.2GPa}}}\end{aligned}\)

04

Calculating For which of the two random samples, the one of part (a) or the one of part (b), is \({\rm{\bar X}}\) more likely to be within \({\rm{1GPa}}\) of \({\rm{70GPa}}\)

Given:

\({\rm{\mu = 70GPa}}\)

\({\rm{\sigma = 1}}{\rm{.6GPa}}\)

Result part (a):

\(\begin{aligned}&{{{\rm{\mu }}_{{\rm{\bar x}}}}{\rm{ = 70GPa}}}\\&{{{\rm{\sigma }}_{{\rm{\bar x}}}}{\rm{ = 0}}{\rm{.4GPa}}}\end{aligned}\)

Result part (b):

\(\begin{aligned}&{{{\rm{\mu }}_{{\rm{\bar x}}}}{\rm{ = 70GPa}}}\\&{{{\rm{\sigma }}_{{\rm{\bar x}}}}{\rm{ = 0}}{\rm{.2GPa}}}\end{aligned}\)

Portion (a) has \({\rm{1GPa}}\) of \({\rm{2}}{\rm{.5}}\) standard deviations, while part (b) has \({\rm{1GPa}}\) of \({\rm{5}}\) standard deviations (b).

If \({\rm{\bar X}}\) has higher standard deviations, it is more likely to be within \({\rm{1GPa}}\) of the mean \({\rm{70GPa}}\), and so it is more likely for part (b).

Note that a smaller standard deviation indicates a narrower distribution, which means that data values are more likely to be closer to the mean.

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