If you want to be 99% confident of estimating the population proportion to within a sampling error of +/- 0.06, what sample size is needed?

Note that the larger the sample size, the narrower is the confidence interval because the margin of error decreases as the sample size is increases. 

Formula for sample size for proportion is calculated as;- n = p_hatch*q_hatch *(Z_score/E) ^ 2

Use excel to get z-score associated with 99% confidence interval, =NORM.S.INV(1-0.01/2)  = 2.58

p_hatch = 0.5 when the proportion is not known.  apply the formula n = 0.5 * 0.5 * (2.58/0.06)^2 = 462.25, the required sample size is 463.

For the second part, we know the population proportion therefore, 0.24 will be used as the population proportion. 

n = 0.24 * 0.76 * (2.58/0.09)^2 =  149.89,  a sample of 150 objects is required to estimate the confidence interval.