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. 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. 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. 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. 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

Solution:

We have that information from the question,

The (mean) of specific type of aluminum alloy sheets is 70 GPa

The (standard deviation) of specific type of aluminum alloy sheets is 1.6 GPa

Consider the Random Sample size n=16 from the question,

The sampling distribution of sample mean for (mean) is

=>

The sampling distribution of sample mean for (standard deviation) is,

From the above calculation, The mean and standard deviation of the sampling distribution of sample mean is 70 GPa and 0.4 GPa

Part a, The mean and standard deviation of the sampling distribution (n=16) of sample mean is 70 GPa and 0.4 GPa

(b)

We have that information,

The (mean) of specific type of aluminum alloy sheets is 70 GPa

The (standard deviation) of specific type of aluminum alloy sheets is 1.6 GPa

Consider the Random Sample size n=256 from the question,

The sampling distribution of sample mean for (mean) is

=>

The sampling distribution of sample mean for (standard deviation) is,

From the above calculation, The mean and standard deviation of the sampling distribution of sample mean is 70 GPa and 0.1 GPa

Part b, The mean and standard deviation of the sampling distribution (n=256) of sample mean is 70 GPa and 0.1 GPa

(c)

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.