使用R中的模拟测试置信区间的覆盖概率

Question

I have a homework question which asks me to test the coverage probability of a confidence interval (found as part of a previous question) using a simulation in R.

My code is attempting to generate 1000 random samples (with replacement) from the sample data I have, effectively treating my original sample as my new population. The random samples are the same size as my population. I then want to find the 95% confidence interval for each random sample and see how many contain the 'true mean' (given in the problem statement) vs. the 'population mean' (the mean of my original sample).

set.seed(1987)

iq <- rnorm(1000,91.08065,14.40393)

pop_mean <- mean(iq) #the mean of my sample is now considered the population mean
true_mean <- 100 #the true mean is 100, specified in question

sampSEs <- numeric() #create an empty vector to put the sample SEs in
sampMeans <- numeric() #create an empty vector to put the sample means in

get_conf_interval <- function(sample_measurements) {
  iqSE_samp <- 15/sqrt(length(iq)) #find the SE using an sd of 15
  iqMean_samp <- mean(sample_measurements) #take the mean of each sample
  upper <- iqMean_samp + 1.96*iqSE_samp #find the upper bound for a 95% CI
  lower <- iqMean_samp - 1.96*iqSE_samp #find the lower bound for a 95% CI
  list(lower=lower, upper=upper)
}

interval_contains_true_mean <- function(interval) { #check if the interval contains the true mean
  true_mean >= interval$lower && true_mean <= interval$upper
}
interval_contains_population_mean <- function(interval) { #check if the interval contains the population mean
  pop_mean >= interval$lower && pop_mean <= interval$upper
}

samples <- replicate(1000, sample(iq, size = 124, replace = T)) #take 1000 samples with replacement from my iq data

for(i in 1:1000) { #for each sample taken
  sampMeans[i] <- mean(samples[i]) #put the mean of it in the vector created previously
  sampSEs[i] <- 15/sqrt(length(iq)) #put the SE in a vector... these are all the same bc not finding the sample sd
}

intervals <- apply(samples, FUN=get_conf_interval, MARGIN=2) #call the function to find the confidence intervals

sampMeans #just check if worked
#sampSEs #ditto

percent_intervals_with_true_mean <- mean(sapply(intervals, FUN=interval_contains_true_mean)) * 100
cat("% Intervals Containing True Mean: ", percent_intervals_with_true_mean, "%\n")

percent_intervals_with_pop_mean <- mean(sapply(intervals, FUN=interval_contains_population_mean)) * 100
cat("% Intervals Containing Population Mean: ", percent_intervals_with_pop_mean, "%")

This code reports that 0% of the confidence intervals for my samples contain the true mean. This is not correct; I have looked at the sample means and several of them ARE the true mean.

Answer 1

1.- i have two solution the first solution is put comma in 'mean(samples[,i])' and

'set.seed(1987)

sigma_M=14.40393

mu_M=91.08065

m=10

iq <- rnorm(m,mu_M,sigma_M)

pop_mean <- mean(iq) #the mean of my sample is now considered the population mean

samples <- replicate(m, sample(iq, size = 4, replace = T)) #take m samples with replacement from my iq data

sampSEs <- numeric() #create an empty vector to put the sample SEs in

sampMeans <- numeric() #create an empty vector to put the sample means in

for(i in 1:m) { #for each sample taken

sampMeans[i] <- mean(samples[,i]) #put the mean of it in the vector created previously

sampSEs[i] <- 15/sqrt(length(iq)) #put the SE in a vector... these are all the same bc not finding the sample sd }

get_conf_interval <- function(sample_measurements) {

iqSE_samp <- 15/sqrt(length(iq)) #find the SE using an sd of 15

iqMean_samp <- mean(sample_measurements) #take the mean of each sample

upper <- iqMean_samp + 1.96*iqSE_samp #find the upper bound for a 95% CI

lower <- iqMean_samp - 1.96*iqSE_samp #find the lower bound for a 95% CI

list(lower=lower, upper=upper) }

interval_contains_population_mean <- function(interval) { #check if the interval contains the population mean

pop_mean >= interval$lower && pop_mean <= interval$upper }

intervals <- apply(samples, FUN=get_conf_interval, MARGIN=2) #call the function to find the confidence intervals

sampMeans #just check if worked

sampSEs #ditto

percent_intervals_with_pop_mean <- mean(sapply(intervals, FUN=interval_contains_population_mean)) * 100

cat("% Intervals Containing Population Mean: ", percent_intervals_with_pop_mean, "%")'

2.- the second solution is change the code but i did only 'pop_mean' population mean (and calculate standard desviation)

'set.seed(1987)

sigma_M=14.40393

mu_M=91.08065

m=10

iq <- rnorm(m,mu_M,sigma_M)

pop_mean <- mean(iq) #the mean of my sample is now considered the population mean

samples <- replicate(m, sample(iq, size = 4, replace = T)) #take m samples with replacement from my iq data

sampMeans=apply(samples, 2, mean)

iqSE_samp <- 15/sqrt(length(iq)) #find the SE using an sd of 15

iqMean_samp <- sampMeans #take the mean of each sample

upper <- iqMean_samp + 1.96*iqSE_samp #find the upper bound for a 95% CI

lower <- iqMean_samp - 1.96*iqSE_samp #find the lower bound for a 95% CI

intervals=cbind(lower, upper)

percent_intervals_with_pop_mean=mean(apply(intervals, 1, findInterval, x = pop_mean) == 1)*100

cat("% Intervals Containing Population Mean: ", percent_intervals_with_pop_mean, "%")'

and the final is 80 porcent for me