Answer:
a) X^ = 70 GPa , s = 0.4 GPa
b) X^ = 70 GPa , s = 0.2 GPa
c) n = 64 .. part b
Step-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 GPa
a)
- 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 GPa
b)
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 GPa
c)
- 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
