## Sample size calculation in clinical trial

Could you please help me understand how did they calculate the sample size in this randomized trial ?

It is a randomized trial comparing the use of hyoscine and placebo on polyp detection rate during colonoscopy.

"Sample size was calculated based on the previous
data as follows; the proportion of patients with
polyps was known to be about 24% (13), and we
assumed that an increased polyp detection rate of
25%, the miss rate from tandem colonoscopy (4, 14),
would be of clinical significance, which yielded 58
patients in each group with a statistical power of
80% and α value 0.05."

I used two different equations but i could not get the 58 they got !!! any help, because it is important for my study.

N = 1/2 X (1.96 + 0.842)2
(arcsin√0.24 - arcsin√0.49)2

This is the other formula i used

N = (1.96 + 0.842)2 * (0.25x0.75 + 0.49x0.51)
(0.25)2

My proportion of patients with polyps are 32%
and i want to power it to detect a difference of 20%, at the same power and a value.

again this is another example (comparing menthol and placebo) but i also did not understand how they did it. and how did they got 220.

"In a previous study of colonoscopy, the adenoma detection rate of white light imaging was 37.5%. For the current study, the sample size was calculated to be sufficient to detect 20% more neoplasms in the L-menthol group than
the placebo group.The power analysis indicated that more than
194 patients were needed in total, 97 per arm, assuming a 5% significance
level and statistical power of 80% using two-sided equivalence. It was therefore estimated that a total of 220 patients would be needed."