# A psychologist is interested in constructing a 99% confidence interval for the proportion of people who accept the theory that a person's spirit is no

A psychologist is interested in constructing a $99\mathrm{%}$ confidence interval for the proportion of people who accept the theory that a persons
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Nathalie Redfern
Step 1
Solution:
Let X be the number of people agreed with the theory and n be the sample number of people.
From the given information, $X=68\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}n=702.$
Step 2
The sample proportion is
$\stackrel{^}{p}=\frac{X}{n}$
$=\frac{68}{702}$
$=0.0969$
Step 3
The confidence level is 0.99.
$1-\alpha =0.99$
$\alpha =1-0.99$
$\alpha =0.01$
Then, the $99\mathrm{%}$ confidence interval for the proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain is
$\stackrel{^}{p}±{Z}_{\frac{\alpha }{2}}\left(\sqrt{\frac{\stackrel{^}{p}\left(1-\stackrel{^}{p}\right)}{n}}\right)=0.0969±{Z}_{\frac{0.01}{2}}\left(\frac{\sqrt{0.0969\left(1-0.0969\right)}}{702}\right)$
$0.0969±{Z}_{0.005}\left(\frac{\sqrt{0.0969\left(1-0.0969\right)}}{702}\right)$
$0.0969±\left(2.58\right)\left(\frac{\sqrt{0.0969\left(1-0.0969\right)}}{702}\right)$ [using the excel function = NORM.INV(0.005, 0, 1)]
$=0.0969±0.0288$
$=\left(0.0969-0.0288,0.0969+0.0288\right)$
$=\left(0.0681,0.1257\right)$
Thus, with $99\mathrm{%}$ confidence the proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain is between 0.0681 and 0.1257.
Step 4
b)
If many groups of 702 randomly selected people are surveyed, then a different confidence interval would be produced from each group. About 99 percent of these confidence intervals will contain the true population proportion of all people who accept the theory that a person’s spirit is no more than the complicated network of neurons in the brain and about $100-99=1$ percent will not contain the true population proportion.