Answer:
The minimum sample size is 1867 observations.
Step-by-step explanation:
We need to construct an 85% confidence interval that has an error less than 0.06. It means that the difference between the upper limit (UL) and the lower limit (LL) has to be 0.06.
[tex]UL-LL= e =0.06[/tex]
[tex]UL-LL= e =0.06\\X+z*s/\sqrt{n} -(X-z*s/\sqrt{n}) = e\\2*z*s/\sqrt{n}=e\\[/tex]
The only variable we can adjust is the number of observations (n)
[tex]2*z*s/\sqrt{n}=e\\\\\sqrt{n}=\frac{2*z*s}{e}\\\\ n=(\frac{2*z*s}{e})^{2}=\frac{4*z^{2} *s^{2} }{e^{2} }[/tex]
For a 85% confidence interval, the z-score is 1.440.
The estimated variance (s^2) is 0.81.
The error e is 0.06.
[tex]n= 4*(1.440)^{2} *0.81/(0.06)^{2} \\\n=4*2.0736*0.81/0.0036= 6.7184 / 0.0036 = 1866.24[/tex]
The sample has to be at least of 1867 observations.
