A 2015 Gallup poll of 1,627 adults found that only 22% felt fully engaged with their mortgage provider. What is the sample size?

Question

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

What is the sample size?

The margin of error for this data set is 2.5%. What is the 95% confidence interval using only 1 decimal?

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Hadley 2 months 2021-10-09T01:14:03+00:00 1 Answer 0 views 0

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    2021-10-09T01:15:54+00:00

    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\%) .

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