Respuesta :
Let X be the length of rulers. X follows Normal distribution with mean μ and standard deviation σ = 0.02. A sample of size n=4 is drawn from population and sample mean m=1000. We have to find 90% confidence interval for population mean μ.
Here population standard deviation is given to us so we will use z confidence interval for mean. It is given as
(Sample mean - Margin of error , Sample mean + Margin of error)
Where margin of error for z confidence interval is
ME = σ [tex] \frac{z_{\alpha/2}}{\sqrt{n}} [/tex]
Where σ = population standard deviation = 0.02
n= sample size =4
[tex] z_{\alpha/2} [/tex] = Critical z score value for given confidence level
We have to find here 90% confidence interval so confidence level c= 0.9
α = 1 -c =1-0.9 = 0.1
[tex] z_{\alpha/2} [/tex] = [tex] z_{0.1/2} [/tex] = [tex] z_{0.05} [/tex]
Here we have to find z score value such that area below it is -z is 0.05 and above z is 0.05
Using excel function to find z score value
=NORM.S.INV(0.05) = -1.645
For calculating confidence interval we consider positive z score value which is 1.645
So the margin of error is
ME = [tex] \frac{0.02 * 1.645}{\sqrt{4}} [/tex]
ME = 0.01645
90% confidence interval for mean is
(Sample mean - ME, Sample mean+ME)
1000- 0.01645, 1000+0.01645
(1000.01645, 999.9836)
Rounding interval upto 3 decimal places
(1000.016, 999.984)
90% confidence interval for population mean is (1000.016, 999.984)
