UkusakazaL

2021-07-30

A paper reported a $\left(1-\alpha \right)$ confidence interval for the proportion of voters is (0.561,0.599) based on a sample of 2,056 people. However, the paper omitted the value of $\alpha$. If you want to test the hypothesis that the proportion of voters is greater than $65\mathrm{%}$ at $1\mathrm{%}$ significance, find ${z}_{calc}$ value for this problem? Please report your answer to 2 decimal places.

### Answer & Explanation

ka1leE

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,
$\stackrel{^}{p}=\frac{0.561+0.599}{2}$
$=0.58$
Length of CI is expressed as difference between given proportion.
Margin of error is expressed as,
Margin of error(E) $=\frac{1}{2}×$ length of CI
$=\frac{1}{2}×\left(0.599-0.561\right)$
$=\frac{1}{2}×0.038$
$=0.019$
Step 2
Hence the CI of the given proportion with margin of error is,
$\stackrel{^}{p}+E=0.58+0.019=0.599$
$\stackrel{^}{p}-E=0.58-0.019=0.561$
Hence, the proportions are (0.599,0.561)
The expression for standard deviation is,
$S.E=\sqrt{\frac{\stackrel{^}{p}\left(1-\stackrel{^}{p}\right)}{n}}$
$=\sqrt{\frac{0.58×0.42}{2056}}$
$=\sqrt{\frac{0.2436}{2056}}$
$=\sqrt{0.00011848}$
$=0.01088$
Step 3
As know that,
Margin of error $\left(E\right)={z}_{\frac{\alpha }{2}}×S.E$
Hence, the value of z is,
${z}_{\frac{\alpha }{2}}=\frac{E}{S.E}$
$=\frac{0.019}{0.01088}$
$\approx 1.7463$
Thus
$\frac{\alpha }{2}=P\left(Z<-1.7463\right)$
$\frac{\alpha }{2}=0.0274$
$\alpha =0.0548$
Hence the confidence level is $\left(1-\alpha \right)=0.9452$.
Step 4
Now, to find null hypothesis:
${H}_{0}:p=65$
${H}_{0}:p>65$
Here, sample proportion is $0.58,n=2056$ and claimed proportion is 0.65 at significance level $\alpha =0.01$.
Hence the standard deviation at 0.65 claimed proportion,
$S.E=\sqrt{\frac{\text{claimed proportion(1-claimed proportion)}}{n}}$
$=\sqrt{\frac{0.65×0.35}{2056}}$
$=\sqrt{\frac{0.2275}{2056}}$
$\approx 0.0105$
So the value of ${z}_{calc}$ is:
${z}_{calc}=\frac{\stackrel{^}{p}-0.65}{0.0105}$
$=\frac{0.58-0.65}{0.0105}$
$=\frac{-0.07}{0.0105}$
$=-6.6667$
Hence, the value of z is -6.67.

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