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 more than the complicated network of neurons in the brain. 68 of the 702 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible. a) With 99% 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 ____ and ____. b) If many groups of 702 randomly selected people are 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 all people who accept the theory th

Question
Confidence intervals
asked 2021-02-25
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 more than the complicated network of neurons in the brain. 68 of the 702 randomly selected people who were surveyed agreed with this theory. Round answers to 4 decimal places where possible.
a)
With \(99\%\) 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 ____ and ____.
b)
If many groups of 702 randomly selected people are 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 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 ____ percent will not contain the true population proportion.

Answers (1)

2021-02-26
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, \(\displaystyle{X}={68}{\quad\text{and}\quad}{n}={702}.\)
Step 2
The sample proportion is
\(\displaystyle\hat{{p}}=\frac{X}{{n}}\)
\(\displaystyle=\frac{68}{{702}}\)
\(\displaystyle={0.0969}\)
Step 3
The confidence level is 0.99.
\(\displaystyle{1}-\alpha={0.99}\)
\(\displaystyle\alpha={1}-{0.99}\)
\(\displaystyle\alpha={0.01}\)
Then, the \(99\%\) 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
\(\displaystyle\hat{{p}}\pm{Z}_{{\frac{\alpha}{{2}}}}{\left(\sqrt{{\frac{{\hat{{p}}{\left({1}-\hat{{p}}\right)}}}{{n}}}}\right)}={0.0969}\pm{Z}_{{\frac{0.01}{{2}}}}{\left(\frac{\sqrt{{{0.0969}{\left({1}-{0.0969}\right)}}}}{{702}}\right)}\)
\(\displaystyle{0.0969}\pm{Z}_{{{0.005}}}{\left(\frac{\sqrt{{{0.0969}{\left({1}-{0.0969}\right)}}}}{{702}}\right)}\)
\(\displaystyle{0.0969}\pm{\left({2.58}\right)}{\left(\frac{\sqrt{{{0.0969}{\left({1}-{0.0969}\right)}}}}{{702}}\right)}\) [using the excel function = NORM.INV(0.005, 0, 1)]
\(\displaystyle={0.0969}\pm{0.0288}\)
\(\displaystyle={\left({0.0969}-{0.0288},{0.0969}+{0.0288}\right)}\)
\(\displaystyle={\left({0.0681},{0.1257}\right)}\)
Thus, with \(99\%\) 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 \(\displaystyle{100}-{99}={1}\) percent will not contain the true population proportion.
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