The confidence interval for population mean is given by :-
[tex]\overline{x}\pm z^* SE[/tex] (1)
, where [tex]\overline{x}[/tex] = sample mean
z* = critical value.
SE = standard error
and [tex]SE=\dfrac{\sigma}{\sqrt{n}}[/tex] , [tex]\sigma[/tex] = population standard deviation.
n= sample size.
As per given , we have
[tex]\overline{x}=74.021[/tex]
[tex]\sigma=0.001[/tex]
n= 15
It is known that ring diameter is normally distributed.
[tex]SE=\dfrac{0.001}{\sqrt{15}}=0.000258198889747\approx0.000258199[/tex]
By z-table ,
The critical value for 95% confidence = z*= 1.96
A 99% two-sided confidence interval on the true mean piston diameter :
[tex]74.021\pm (2.576) (0.000258199)[/tex] (using (1))
[tex]74.021\pm 0.000665120624[/tex]
[tex]74.021\pm 0.000665120624\\\\=(74.021- 0.000665120624,\ 74.021+ 0.000665120624)\\\\=(74.0203348794,\ 74.0216651206)\approx(74.020,\ 74.022)[/tex] [Rounded to three decimal places]
∴ A 99% two-sided confidence interval on the true mean piston diameter = (74.020, 74.022)
By z-table ,
The critical value for 95% confidence = z*= 1.96
A 95% lower confidence bound on the true mean piston diameter:
[tex]74.021- (1.96) (0.000258199)[/tex] (using (1))
[tex]74.021- 0.00050607004=74.02049393\approx74.020[/tex] [Rounded to three decimal places]
∴ A 95% lower confidence bound on the true mean piston diameter= 74.020
