Question

QUESTION # 1

A random sample of 5,280 permanent dwellings on an entire reservation showed that 1,634 were traditional hogans.

(a)Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find a point estimate for p. (Round your answer to four decimal places.)

(b)Find a 99% confidence interval for p. (Round your answer to three decimal places.)

lower limit

upper limit

QUESTION #2

A random sample of 5,100 permanent dwellings on an entire reservation showed that 1,690 were traditional hogans.

(a)Let p be the proportion of all permanent dwellings on the entire reservation that are traditional hogans. Find point estimates for p and q. (Round your answer to four decimal places.)

p̂=

q̂=

(b)Find a 99% confidence interval for p.

Find the maximal margin of error, E. (Round your answer to three decimal places.)

E =

Report the bounds from the 95% confidence interval for p. (Round your answers to three decimal places.)

lower limit

upper limit

Answer 1

Solution :

Given that,

1) n = 5280

x = 1634

a) Point estimate = sample proportion = $\hat p$ = x / n = 1634 / 5280 = 0.3095

1 - $\hat p$ = 1 - 0.3095 = 0.6905

b) At 99% confidence level

$\alpha$ = 1 - 99%

$\alpha$ =1 - 0.99 =0.01

$\alpha$/2 = 0.005

Z$\alpha$/2 = Z_{0.005 = 2.576}

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 2.576 ($\sqrt$((0.3095 * 0.6905) / 5280)

= 0.016

A 99% confidence interval for population proportion p is ,

$\hat p$ ± E  

= 0.3095  ± 0.016

= ( 0.294, 0.326 )

lower limit = 0.294

upper limit = 0.326

2) Given that,

n = 5100

x = 1690

a) Point estimate = sample proportion = $\hat p$ = x / n = 1690 / 5100 = 0.3314

q̂= 1 - $\hat p$ = 1 - 0.3314 = 0.6686

b) At 99% confidence level

$\alpha$ = 1 - 99%

$\alpha$ =1 - 0.99 =0.01

$\alpha$/2 = 0.005

Z$\alpha$/2 = Z_{0.005 = 2.576}

Margin of error = E = Z_{$\alpha$ / 2} * $\sqrt$ (($\hat p$ * (1 - $\hat p$ )) / n)

= 2.576 ($\sqrt$((0.3314 * 0.6686) / 5100)

= 0.017

A 99% confidence interval for population proportion p is ,

$\hat p$ ± E

= 0.3314 ± 0.017

= ( 0.314, 0.348 )

lower limit = 0.314

upper limit = 0.348