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.

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.