Given data,

Total number of people is 2056.

The proportion of voters is (0.561, 0.599)

Step 1

This is a symmetrical CI.

Hence the sample proportion is expressed as,

\(\displaystyle\hat{{{p}}}={\frac{{{0.561}+{0.599}}}{{{2}}}}\)

\(\displaystyle={0.58}\)

Length of CI is expressed as difference between given proportion.

Margin of error is expressed as,

Margin of error(E) \(\displaystyle={\frac{{{1}}}{{{2}}}}\times\) length of CI

\(\displaystyle={\frac{{{1}}}{{{2}}}}\times{\left({0.599}-{0.561}\right)}\)

\(\displaystyle={\frac{{{1}}}{{{2}}}}\times{0.038}\)

\(\displaystyle={0.019}\)

Step 2

Hence the CI of the given proportion with margin of error is,

\(\displaystyle\hat{{{p}}}+{E}={0.58}+{0.019}={0.599}\)

\(\displaystyle\hat{{{p}}}-{E}={0.58}-{0.019}={0.561}\)

Hence, the proportions are (0.599,0.561)

The expression for standard deviation is,

\(\displaystyle{S}.{E}=\sqrt{{{\frac{{\hat{{{p}}}{\left({1}-\hat{{{p}}}\right)}}}{{{n}}}}}}\)

\(\displaystyle=\sqrt{{{\frac{{{0.58}\times{0.42}}}{{{2056}}}}}}\)

\(\displaystyle=\sqrt{{{\frac{{{0.2436}}}{{{2056}}}}}}\)

\(\displaystyle=\sqrt{{{0.00011848}}}\)

\(\displaystyle={0.01088}\)

Step 3

As know that,

Margin of error \(\displaystyle{\left({E}\right)}={z}_{{\frac{\alpha}{{2}}}}\times{S}.{E}\)

Hence, the value of z is,

\(\displaystyle{z}_{{\frac{\alpha}{{2}}}}={\frac{{{E}}}{{{S}.{E}}}}\)

\(\displaystyle={\frac{{{0.019}}}{{{0.01088}}}}\)

\(\displaystyle\approx{1.7463}\)

Thus

\(\displaystyle\frac{\alpha}{{2}}={P}{\left({Z}{<}-{1.7463}\right)}\)

\(\displaystyle\frac{\alpha}{{2}}={0.0274}\)

\(\displaystyle\alpha={0.0548}\)

Hence the confidence level is \(\displaystyle{\left({1}−\alpha\right)}={0.9452}\).

Step 4

Now, to find null hypothesis:

\(\displaystyle{H}_{{{0}}}:{p}={65}\)

\(\displaystyle{H}_{{{0}}}:{p}{>}{65}\)

Here, sample proportion is \(\displaystyle{0.58},{n}={2056}\) and claimed proportion is 0.65 at significance level \(\displaystyle\alpha={0.01}\).

Hence the standard deviation at 0.65 claimed proportion,

\(\displaystyle{S}.{E}=\sqrt{{{\frac{{\text{claimed proportion(1-claimed proportion)}}}{{{n}}}}}}\)

\(\displaystyle=\sqrt{{{\frac{{{0.65}\times{0.35}}}{{{2056}}}}}}\)

\(\displaystyle=\sqrt{{{\frac{{{0.2275}}}{{{2056}}}}}}\)

\(\displaystyle\approx{0.0105}\)

So the value of \(\displaystyle{z}_{{{c}{a}{l}{c}}}\) is:

\(\displaystyle{z}_{{{c}{a}{l}{c}}}={\frac{{\hat{{{p}}}-{0.65}}}{{{0.0105}}}}\)

\(\displaystyle={\frac{{{0.58}-{0.65}}}{{{0.0105}}}}\)

\(\displaystyle={\frac{{-{0.07}}}{{{0.0105}}}}\)

\(\displaystyle=-{6.6667}\)

Hence, the value of z is -6.67.