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?
Accepted Solution
A:
Answer:a) X^ = 70 GPa , s = 0.4 GPab) X^ = 70 GPa , s = 0.2 GPac) n = 64 .. part bStep-by-step explanation:Solution:-- A sample ( n ) was taken from aluminum alloy sheets of a particular type the distribution parameters are given below: Mean ( u ) = 70 GPa Standard deviation ( σ ) = 1.6 GPaa)- We take a sample size of n = 16. The random variable X denotes the distribution of the sample obtained.- We will estimate the parameters of the sample distribution X.- The point estimate method tells us that the population mean ( u ) is assumed as the sample mean ( X^ ). Sample Mean ( X^ ) = u = 70 GPa- The sample standard deviation ( s ) for the given sample with known population standard deviation ( σ ) is given by: sample standard deviation ( s ) = σ / √n sample standard deviation ( s ) = 1.6 / √16 sample standard deviation ( s ) = 0.4 GPab) Repeat the above calculations for sample size n = 64.- We will estimate the parameters of the sample distribution X.- The point estimate method tells us that the population mean ( u ) is assumed as the sample mean ( X^ ). Sample Mean ( X^ ) = u = 70 GPa- The sample standard deviation ( s ) for the given sample with known population standard deviation ( σ ) is given by: sample standard deviation ( s ) = σ / √n sample standard deviation ( s ) = 1.6 / √64 sample standard deviation ( s ) = 0.2 GPac)- The standard deviation ( s ) gives us the uncertainty of mean ( X^ ). How spread apart/close are the data points from the mean. - We see that standard deviation ( s ) has an inverse relation to the sample size ( n ): sample standard deviation ( s ) = σ / √n- So with increasing sample size the there is a decreased variability in the sample distribution of ( X ).Answer: The sample size n = 64 used in part b would give us lesser variability of sample distribution of X as compared to sample size n = 16 used in part a. Hence, X is more likely to be within 1 GPa of the mean in part (b). This is due to the decreased variability