Question

A 2015 Gallup poll of 1,627 adults found that only 22% felt fully engaged with their mortgage provider.

Answer 1

Answer: $(20.0\%,24.0\%)$

Step-by-step explanation:

Given, A 2015 Gallup poll of 1,627 adults found that only 22% felt fully engaged with their mortgage provider.

Here , 1,627 adults are determining the sample.

Thus, sample size : n = 1627

Also, the sample proportion of adults felt fully engaged with their mortgage provider are $\hat{p}=22\%=0.22$

for 95% confidence level , critical z-value =1.96

Then , the 95% confidence interval would be :-

$\hat{p}\pm z\sqrt{\dfrac{\hat{p}(1-\hat{p})}{n}}$

Substituting values , we get

$0.22\pm 1.96(\sqrt{\dfrac{0.22(1-0.22)}{1627}})\\\\=0.22\pm 1.96\sqrt{\dfrac{0.22\times0.78}{1627}}\\\\\approx0.22\pm0.0201\\\\=(0.22-0.0201,\ 0.22+0.0201)\\\\=(0.1999,\ 0.2401)\approx(20.0\%,24.0\%)$

Hence, the required 95% confidence interval using only 1 decimal : $(20.0\%,24.0\%)$ .