Respuesta :
Answer:
(a) Margin of error ( E) = $2,000 , n = 54
(b) Margin of error ( E) = $1,000 , n = 216
(c) Margin of error ( E) = $500 , n= 864
Step-by-step explanation:
Given -
Standard deviation [tex]\sigma[/tex] = $7,500
[tex]\alpha[/tex] = 1 - confidence interval = 1 - .95 = .05
[tex]Z_{\frac{\alpha}{2}}[/tex] = [tex]Z_{\frac{.05}{2}}[/tex] = 1.96
let sample size is n
(a) Margin of error ( E) = $2,000
Margin of error ( E) = [tex]Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}[/tex]
E = [tex]Z_{\frac{.05}{2}}\frac{7500}{\sqrt{n}}[/tex]
Squaring both side
[tex]E^{2} = 1.96^{2}\times\frac{7500^{2}}{n}[/tex]
[tex]n =\frac{1.96^{2}}{2000^{2}} \times 7500^{2}[/tex]
n = 54.0225
n = 54 ( approximately)
(b) Margin of error ( E) = $1,000
E = [tex]Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}[/tex]
1000 = [tex]Z_{\frac{.05}{2}}\frac{7500}{\sqrt{n}}[/tex]
Squaring both side
[tex]1000^{2} = 1.96^{2}\times\frac{7500^{2}}{n}[/tex]
[tex]n =\frac{1.96^{2}}{1000^{2}} \times 7500^{2}[/tex]
n = 216
(c) Margin of error ( E) = $500
E = [tex]Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt{n}}[/tex]
500 = [tex]Z_{\frac{.05}{2}}\frac{7500}{\sqrt{n}}[/tex]
Squaring both side
[tex]500^{2} = 1.96^{2}\times\frac{7500^{2}}{n}[/tex]
[tex]n =\frac{1.96^{2}}{500^{2}} \times 7500^{2}[/tex]
n = 864
