Step 1

Given:

Margin of error (E) = 0.02

Confidence level = 95%

Population proportion (p) = 0.26

Step 2

(a)

The sample size could be calculated using the following formula:

\(\displaystyle{n}={p}{\left({1}-{p}\right)}{\left(\frac{{Z}_{{\frac{\alpha}{{2}}}}}{{E}}\right)}^{{2}}\)

Step 3

The sample size at 95% confidence level can be calculated as:

The value of \(\displaystyle\frac{{Z}_{\alpha}}{{2}}\) corresponding to 95% confidence level is 1.96

\(\displaystyle{n}={p}{\left({1}-{p}\right)}{\left(\frac{{Z}_{{\frac{\alpha}{{2}}}}}{{E}}\right)}^{{2}}\)

\(\displaystyle={0.26}{\left({1}-{0.26}\right)}{\left(\frac{{1.96}}{{0.02}}\right)}^{{2}}\)

\(\displaystyle={0.1924}\times{9604}\)

=1847.810

\(\displaystyle\approx{1848}\)

Step 4

(b)

Here, X = 520 and n = 1848.

The point estimate can be calculated as:

\(\displaystyle\hat{{{p}}}=\frac{{X}}{{n}}\)

\(\displaystyle=\frac{{520}}{{1848}}\)

=0.2814

Step 5

(c)

The 95% confidence interval for the proportion of smokers in the population can be calculated as:

\(\displaystyle{C}{I}=\hat{{{p}}}\pm{Z}_{{\frac{\alpha}{{2}}}}\sqrt{{\frac{{\hat{{{p}}}{\left({1}-\hat{{{p}}}\right)}}}{{n}}}}\)

\(\displaystyle={0.2814}\pm{1.96}\times\sqrt{{\frac{{{0.2814}{\left({1}-{0.2814}\right)}}}{{1848}}}}\)

\(\displaystyle={0.2814}\pm{0.0205}\)

=(0.2609,0.3019)

Given:

Margin of error (E) = 0.02

Confidence level = 95%

Population proportion (p) = 0.26

Step 2

(a)

The sample size could be calculated using the following formula:

\(\displaystyle{n}={p}{\left({1}-{p}\right)}{\left(\frac{{Z}_{{\frac{\alpha}{{2}}}}}{{E}}\right)}^{{2}}\)

Step 3

The sample size at 95% confidence level can be calculated as:

The value of \(\displaystyle\frac{{Z}_{\alpha}}{{2}}\) corresponding to 95% confidence level is 1.96

\(\displaystyle{n}={p}{\left({1}-{p}\right)}{\left(\frac{{Z}_{{\frac{\alpha}{{2}}}}}{{E}}\right)}^{{2}}\)

\(\displaystyle={0.26}{\left({1}-{0.26}\right)}{\left(\frac{{1.96}}{{0.02}}\right)}^{{2}}\)

\(\displaystyle={0.1924}\times{9604}\)

=1847.810

\(\displaystyle\approx{1848}\)

Step 4

(b)

Here, X = 520 and n = 1848.

The point estimate can be calculated as:

\(\displaystyle\hat{{{p}}}=\frac{{X}}{{n}}\)

\(\displaystyle=\frac{{520}}{{1848}}\)

=0.2814

Step 5

(c)

The 95% confidence interval for the proportion of smokers in the population can be calculated as:

\(\displaystyle{C}{I}=\hat{{{p}}}\pm{Z}_{{\frac{\alpha}{{2}}}}\sqrt{{\frac{{\hat{{{p}}}{\left({1}-\hat{{{p}}}\right)}}}{{n}}}}\)

\(\displaystyle={0.2814}\pm{1.96}\times\sqrt{{\frac{{{0.2814}{\left({1}-{0.2814}\right)}}}{{1848}}}}\)

\(\displaystyle={0.2814}\pm{0.0205}\)

=(0.2609,0.3019)