Question

In a recent poll of 1,200 homeowners in the United States, one in five homeowners reports having a home equity loan that he or she is currently paying off. Using a confidence coefficient of 0.99, derive the interval estimate for the proportion of all homeowners in the United States that hold a home equity loan.

Answer 1

Answer:

Interval estimate for the proportion of all homeowners in the United States that hold a home equity loan lie between (.17033 , .22967 ) or 17% to 22.9%

Step-by-step explanation:

Given -

In a recent poll of 1,200 homeowners in the United States, one in five homeowners reports having a home equity loan that he or she is currently paying off.

If one in five homeowners reports having a home equity loan then for 1200 homeowners = $\frac{1200}{5}$ = 240 homeowners reports having a home equity loan.

Sample proportion $(\widehat{p})$ = $\frac{240}{ 1200}$ = 0.2

confidence coefficient = 0.99

$(\alpha)$ = 1 - confidence coefficient  = 1 - 0.99 = .01

$z_{\frac{\alpha}{2}} = z_{\frac{.01}{2}}$  =  2.58

interval estimate for the proportion of all homeowners in the United States that hold a home equity loan

=  $\widehat{p}\pm z_{\frac{\alpha}{2}}\sqrt\frac{{\widehat{p}( 1 - \widehat{p})}}{n}$

     =  $0.2\pm z_{\frac{.01}{2}}\sqrt\frac{{0.2( 1 - 0.2)}}{1200}$

    = $0.2\pm 2.58\times \sqrt\frac{{0.2( 0.8)}}{1200}$

    = $0.2\pm .02967$

    = $(0.2 - .02967 ) , (0.2 + .02967)$

    = (.17033 , .22967 )