Confidence intervals, again Several factors are involved in the creat

Step 1
(a) As the confidence level increases, we need to be more sure that the confidence interval contains the true population proportion and thus the confidence interval needs to contain more possible values, which means that the confidence interval be wider.
Since the margin of error is half the width of the confidence interval, the margin of error then needs to increase (not decrease) and thus the given statement is false.
Step 2
(b) As the sample larger, we have more information about the population proportion and thus our estimates will be more accurate.
The more accurate our estimates are, the $n\Rightarrow er$ the confidence interval.
Since the margin of error is half the width of the confidence interval, the margin of error then needs to decrease and thus the given statement is true.
Step 3
(c) As the sample becomes larges, we have more information about the population proportion and thus our estimates will be more accurate.
The more accurate our estimates are, the more confident we are that the population proportion has a certain margin of error and thus the confidence level increases, which means that the statement is true.
Step 4
(d) Formula for the margin of error:
${\displaystyle E=z_{\frac{\alpha }{2}}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}}$
When the margin of error is halved (divided by 2):
${\displaystyle \frac{E}{2}=z_{\frac{\alpha }{2}}\cdot \frac{1}{2}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{n}}=z_{\frac{\alpha }{2}}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{2^{2}n}}=z_{\frac{\alpha }{2}}\cdot \sqrt{\frac{\hat{p}(1-\hat{p})}{4n}}}$
We then note that n is replaced by 4n and thus the sample size quadruples (not doubles) as the margin of error halves and thus the given statement is false.

Step-by-step explanation:
Confidence interval is one which has centre as mean.  The width of the interval would be 2 times margin of error.
$\text{Margin of error}=\text{Critical value}\cdot \text{Std error/sq rt of sample}$
Hence if margin of error is lower for same n, then it means lower confidence level. Option a is right
Similarly, if bigger the sample size, lower is margin of error. Hence option b is right
c) False.  Smaller samples make margin of error big and hence confidence intervals bigger.
d) If n becomes 9n, then margin of error becomes $\frac{1}{3}$ Hence option d is right.