# A newsgroup is interested in constructing a 90% confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of

A newsgroup is interested in constructing a $90\mathrm{%}$ confidence interval for the proportion of all Americans who are in favor of a new Green initiative. Of the 555 randomly selected Americans surveyed, 437 were in favor of the initiative. Round answers to 4 decimal places where possible.
a)
With $90\mathrm{%}$ confidence the proportion of all Americans who favor the new Green initiative is between ____ and ____.
b)
If many groups of 555 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group.
About ____ percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about ____ percent will not contain the true population proportion.
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Step 1
Consider,
Number of Americans surveyed $\left(n\right)=555$
Number of Americans that are in favor of the initiative $\left(x\right)=437$
(a)
The sample proportion is computed as:
Sample proportion $\left(\stackrel{^}{p}\right)=\frac{x}{n}$
$=\frac{437}{555}$
$=0.7874$
Step 2
From standard normal table, the z-score at $90\mathrm{%}$ confidence level is 1.645.
The $90\mathrm{%}$ confidence interval is computed as:
$CI=\stackrel{^}{p}±{z}_{\alpha \text{/}2}×\sqrt{\frac{\stackrel{^}{p}\left(1-\stackrel{^}{p}\right)}{n}}$
$=0.7874±1.645×\sqrt{\frac{\left(0.7874\right)\left(1-0.7874\right)}{555}}$
$=0.7874±0.0286$
$=\left(0.7588,0.8160\right)$
Therefore, the proportion of Americans that favor the new Green Initiative is between 0.7588 and 0.8160.
Step 3
(b)
The confidence interval provides the range in which the population proportion is likely to fall in. In this case, there is $90\mathrm{%}$ probability that the proportion of Americans that favor the new Green initiative is between 0.7588 and 0.8160.
If many groups of 555 randomly selected Americans were surveyed, then a different confidence interval would be produced from each group. About $90\mathrm{%}$ percent of these confidence intervals will contain the true population proportion of Americans who favor the Green initiative and about $10\mathrm{%}$ percent will not contain the true population proportion.