dplyr.All operations in points 3-10 should be done with dplyr functions 
dat <- read.csv("http://math-info.hse.ru/f/2018-19/pep/hw/CPDS.csv")
# str(dat)
read.csv():dec = "," so that R can regard numbers with commas as numbers, not stringsstringsAsFactors = FALSE so that R can regard text values as character ones, not factor.
Make sure that now everything is correct.
dat <- read.csv("http://math-info.hse.ru/f/2018-19/pep/hw/CPDS.csv",
                dec = ",", stringsAsFactors = TRUE)
year, country, iso, poco, eu, gov_right1, gov_cent1, gov_left1, gov_party, gov_type, womenpar, pop and save them to the data frame small.library(dplyr)
small <- dat %>% select(year, country, iso, poco, eu, 
              gov_right1, gov_cent1, gov_left1, 
              gov_party, gov_type, womenpar, pop)
log_pop with values of the natural logarithm of population and add it to small.small <- dat %>% mutate(log_pop = log(pop))
small correspond to post-communist and not post-communist states?small %>% group_by(poco) %>% tally
## # A tibble: 2 x 2
##    poco     n
##   <int> <int>
## 1     0  1371
## 2     1   279
small? Hint: n_distinct() in dplyr combined with summarise() might be helpful.small %>% group_by(poco) %>% summarise(n = n_distinct(country))
## # A tibble: 2 x 2
##    poco     n
##   <int> <int>
## 1     0    25
## 2     1    11
small %>% summarise(left = mean(gov_left1, na.rm = TRUE),
                                   center = mean(gov_cent1, na.rm = TRUE),
                                   right = mean(gov_right1, na.rm = TRUE))
##       left   center   right
## 1 32.37808 23.45062 39.4144
small %>% group_by(eu) %>% summarise(left = mean(gov_left1, na.rm = TRUE),
                                   center = mean(gov_cent1, na.rm = TRUE),
                                   right = mean(gov_right1, na.rm = TRUE))
## # A tibble: 2 x 4
##      eu  left center right
##   <int> <dbl>  <dbl> <dbl>
## 1     0  30.4   21.0  44.3
## 2     1  34.5   26.0  34.2
small %>% filter(poco == 1 & gov_right1 > 50) %>% View
# rows (observations)
small %>% filter(poco == 1 & gov_right1 > 50) %>% tally
##     n
## 1 126
# countries (unique)
small %>% filter(poco == 1 & gov_right1 > 50) %>% 
  summarise(n = n_distinct(country))
##    n
## 1 10
# calculate a Pearson's correlation coef and test its significance
cor.test(small$womenpar, small$gov_right1)
# save results in test
test <- cor.test(small$womenpar, small$gov_right1)
# look at this structure
str(test)
Now you can choose any element from test, for example, the correlation coefficient itself or corresponding p-value.
test$estimate
test$p.value
small in the following way: calculate the correlation coefficient between womenpar and gov_right1 for post-communist and not post-communist states separately, and report the coefficient and the corresponding p-value for each group.small %>% group_by(poco) %>% 
  summarise(corr = cor.test(womenpar, gov_right1)$estimate,
            pvalue = cor.test(womenpar, gov_right1)$p.value)
## # A tibble: 2 x 3
##    poco   corr     pvalue
##   <int>  <dbl>      <dbl>
## 1     0 -0.121 0.00000703
## 2     1  0.154 0.0108