Question

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 σ X = GPa

(b) Answer the questions posed in part (a) for a sample size of n = 256 sheets. E(X) = GPa σ 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.

1. X 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.

2. 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.

3. X 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.

4. X 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.

Answer 1

(a)

By Central limit theorem, the sampling distribution of mean will be Normal distribution with

mean E(X) = = 70 GPa

and Standard deviation = = 1.6 / = 0.4 GPa

(b)

For n = 256 sheets, the sampling distribution of mean will be Normal distribution with

mean E(X) = = 70 GPa

and Standard deviation = = 1.6 / = 0.1 GPa

(c)

Since the standard deviation of sample mean decreases as the sample size increases, X will more likely to be within 1 GPa of 70 GPa for larger sample size.

1. X 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.