Question

What is the confidence level of each of the following confidence intervals for the population mean μ?

i) x̄ ±1.96(σ/_$\sqrt{n}$)

ii) x̄±1.645(σ/_$\sqrt{n}$)

iii) x̄±2.575(σ/_$\sqrt{n}$)

iv) x̄± 1.28(σ/_$\sqrt{n}$)

v)  x̄±0.99(σ/_$\sqrt{n}$)

Thank You!

Answer 1

SOLUTION:

i) x̄ ±1.96(σ/_$\sqrt{n}$)

At 95% confidence level

$\alpha$ = 1 - 95%  

$\alpha$ = 1 - 0.95 =0.05

$\alpha$/2 = 0.025

Z$\alpha$/2 = Z_{0.025 = 1.96}

_II)ii) x̄±1.645(σ/_$\sqrt{n}$)

At 90% confidence level

$\alpha$ = 1 - 90%  

$\alpha$ = 1 - 0.90 =0.10

$\alpha$/2 = 0.05

Z$\alpha$/2 = Z_{0.05 = 1.645}

₃₎

x̄±2.575(σ/_$\sqrt{n}$)

At 99% confidence level the z is ,

$\alpha$  = 1 - 99% = 1 - 0.99 = 0.01

$\alpha$ / 2 = 0.01 / 2 = 0.005

Z_$\alpha$/2 = Z_0.005 = 2.576

4)x̄± 1.28(σ/_$\sqrt{n}$)

Z_$\alpha$/2 = Z_0.900 = 1.28

5)

x̄±0.99(σ/_$\sqrt{n}$)

Z_$\alpha$/2 = Z_0.839 = 0.99