A paper reported a (1−α) confidence interval for the proportion of voters is (0.561,0.599) based...

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
${\displaystyle =\frac{1}{2}\times (0.599-0.561)}$
${\displaystyle =\frac{1}{2}\times 0.038}$
${\displaystyle =0.019}$
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,
${\displaystyle S.E=\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}$
${\displaystyle =\sqrt{\frac{0.58\times 0.42}{2056}}}$
${\displaystyle =\sqrt{\frac{0.2436}{2056}}}$
${\displaystyle =\sqrt{0.00011848}}$
${\displaystyle =0.01088}$
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
${\displaystyle \frac{\alpha }{2}=P(Z<-1.7463)}$
${\displaystyle \frac{\alpha }{2}=0.0274}$
${\displaystyle \alpha =0.0548}$
Hence the confidence level is ${\displaystyle (1-\alpha )=0.9452}$.
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 ${\displaystyle z_{calc}}$ is:
${\displaystyle z_{calc}=\frac{\hat{p}-0.65}{0.0105}}$
${\displaystyle =\frac{0.58-0.65}{0.0105}}$
${\displaystyle =\frac{-0.07}{0.0105}}$
${\displaystyle =-6.6667}$
Hence, the value of z is -6.67.