a. The confidence level increases as the margin of error increases and vice-versa.

Therefore, the statement "For a given sample size, reducing the margin of error will mean lower confidence' is true.

b. Margin of error:

The margin of error for the estimate of p is,

\(\displaystyle{M}{E}={z}\cdot\sqrt{{{\frac{{\hat{{{p}}}{\left({1}-\hat{{{p}}}\right)}}}{{{n}}}}}}\)

The margin of error decreases as the sample size increases because the margin of error and the sample size are inversely proportional.

Hence, the variability would be less for larger sample which implies to smaller standard deviation that implies smaller margin of error.

Therefore, the statement "For a certain confidence level, you can get a smaller margin of error by selecting a bigger sample" is true.

Step 2

c. For small sample, the variability would be more and the confidence interval becomes less successful in taking the true population proportion.

Therefore, the statement "For a fixed margin of error, smaller samples will mean lower confidence" is true.

d. The margin of error decreases as the sample size decreases because the margin of error and the sample size are inversely proportional.

Therefore, the statement "For a given confidence level, a sample 9 times as large will make a margin of error one third as big" is true.