Question 1

library(pwr)
## Warning: package 'pwr' was built under R version 4.0.5
pwr.t.test(n=NULL,d=-0.5,sig.level=0.05,power=.25,type="two.sample")
## 
##      Two-sample t test power calculation 
## 
##               n = 14.19047
##               d = 0.5
##       sig.level = 0.05
##           power = 0.25
##     alternative = two.sided
## 
## NOTE: n is number in *each* group

The number of samples needed is 15.

Question 2

pwr.t.test(n=NULL,d=0.5,sig.level=0.10,power=.85,type="paired")
## 
##      Paired t test power calculation 
## 
##               n = 30.15891
##               d = 0.5
##       sig.level = 0.1
##           power = 0.85
##     alternative = two.sided
## 
## NOTE: n is number of *pairs*

The number of samples needed in each group is 31 samples, so 62 samples in total.

library(pwr)
pwr.t.test(n=NULL,d=.5,sig.level=0.05,power=.25,type="two.sample")
pwr.t.test(n=NULL,d=.5,sig.level=0.10,power=.85,type="paired")