   Chapter 5.4, Problem 46E

Chapter
Section
Textbook Problem

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? b. Answer the questions posed in part (a) for a sample size of n = 64 sheets. 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.

a.

To determine

Find the location where the sampling distribution of X¯ is centered and the standard deviation of the X¯ distribution for n=16.

Explanation

Given info:

The mean value of Young’s modulus- a measure of stiffness, for a sheet of an aluminum alloy is 70 GPa and the standard deviation is 1.6 GPa. The sample mean for Young’s modulus for a sample of size n is denotes by X¯.

Calculation:

Let the population mean of a random variable X, be μ and the corresponding standard deviation be σ.

It is known that, for a sample of size n, the sample mean X¯ has mean E(X¯) and standard deviation σX¯ are given as:

E(X¯)=μ,

σX¯=σn.

Here, n=16, μ=70, σ=1

b.

To determine

Find the location where the sampling distribution of X¯ is centered and the standard deviation of the X¯ distribution for n=64.

c.

To determine

Explain which of the random samples in parts a and b is more likely to have X¯ within 1 GPa of 70 GPa.

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