Answer:
Option b - not significantly greater than 75%.
Step-by-step explanation:
A random sample of 100 people was taken i.e. n=100
Eighty of the people in the sample favored Candidate i.e. x=80
We have used single sample proportion test,
![p=\frac{x}{n}](https://tex.z-dn.net/?f=p%3D%5Cfrac%7Bx%7D%7Bn%7D)
![p=\frac{80}{100}](https://tex.z-dn.net/?f=p%3D%5Cfrac%7B80%7D%7B100%7D)
![p=0.8](https://tex.z-dn.net/?f=p%3D0.8)
Now we define hypothesis,
Null hypothesis
: candidate A is significantly greater than 75%.
Alternative hypothesis
: candidate A is not significantly greater than 75%.
Level of significance ![\alpha=0.05](https://tex.z-dn.net/?f=%5Calpha%3D0.05)
Applying test statistic Z -proportion,
![Z=\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}](https://tex.z-dn.net/?f=Z%3D%5Cfrac%7B%5Cwidehat%7Bp%7D-p_0%7D%7B%5Csqrt%7B%5Cfrac%7Bp_0%281-p_0%29%7D%7Bn%7D%7D%7D)
Where,
and ![p=75%=0.75](https://tex.z-dn.net/?f=p%3D75%25%3D0.75)
Substitute the values,
![Z=\frac{0.80-0.75}{\sqrt{\frac{0.75(1-0.75)}{100}}}](https://tex.z-dn.net/?f=Z%3D%5Cfrac%7B0.80-0.75%7D%7B%5Csqrt%7B%5Cfrac%7B0.75%281-0.75%29%7D%7B100%7D%7D%7D)
![Z=\frac{0.80-0.75}{\sqrt{\frac{0.1875}{100}}}](https://tex.z-dn.net/?f=Z%3D%5Cfrac%7B0.80-0.75%7D%7B%5Csqrt%7B%5Cfrac%7B0.1875%7D%7B100%7D%7D%7D)
![Z=\frac{0.05}{0.0433}](https://tex.z-dn.net/?f=Z%3D%5Cfrac%7B0.05%7D%7B0.0433%7D)
![Z=1.1547](https://tex.z-dn.net/?f=Z%3D1.1547)
The p-value is
![P(Z>1.1547)=1-P(Z](https://tex.z-dn.net/?f=P%28Z%3E1.1547%29%3D1-P%28Z%3C1.1547%29)
![P(Z>1.1547)=1-0.8789](https://tex.z-dn.net/?f=P%28Z%3E1.1547%29%3D1-0.8789)
![P(Z>1.1547)=0.1241](https://tex.z-dn.net/?f=P%28Z%3E1.1547%29%3D0.1241)
Now, the p-value is greater than the 0.05.
So we fail to reject the null hypothesis and conclude that the A is not significantly greater than 75%.
Therefore, Option b is correct.