
  # verify that the alternative estimator is unbiased
  n <- 100
  w <- c(rep((1+0.5)/n, n/2), rep((1-0.5)/n, n/2))
           
           
  # define the alternative estimator mu_tilde
           
           
  # compute repeatedly estimates for both estimators and store the results in est_bar and est_tilde
  set.seed(123)
           
           
           
  # compute the sample variances for est_bar and est_tilde


           # verify that the alternative estimator is unbiased
           n <- 100
           w <- c(rep((1+0.5)/n, n/2), rep((1-0.5)/n, n/2))
           sum(w)
           
           # define the alternative estimator mu_tilde
           mu_tilde <- function(x){sum(w*x)}
           
           # compute repeatedly estimates for both estimators and store the results in est_bar and est_tilde
           set.seed(123)
           est_bar <- replicate(expr = mean(rnorm(100, 5, 10)), n = 10000)
           est_tilde <- replicate(expr = mu_tilde(rnorm(100, 5, 10)), n = 10000)
           
           # compute the sample variances for est_bar and est_tilde
           var(est_bar)
           var(est_tilde)


           test_function_result("sum")
           test_function_definition("mu_tilde",
           function_test = {
           test_expression_result("mu_tilde(1:100)")
           test_expression_result("mu_tilde(2:101)")
           })
           test_object("est_bar")
           test_object("est_tilde")
           test_function_result("var", index = 1)
           test_function_result("var", index = 2)
           success_msg("Correct! The sample mean is more efficient (that is, has a lower variance) than the alternative estimator.")