USE R - STUDIO TO SOLVE. What proportion of times did the confidence intervals m
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Question
USE R - STUDIO TO SOLVE.
What proportion of times did the confidence intervals miss the true mean? using a simulation to answer the question. In your simulation, draw N=10^5 random samples (of size n=20) from the right-skewed Gamma distribution Gamma(5, 2) with parameters r=5 and lambda=2. Use also q = qt(0.975, n-1) to determine Upper and Lower limit of the interval. Count the number of times the 95% confidence interval misses the mean mu=5/2 on each side. Repeat this simulation by changing the sample size, n=30, n=60, n=100, and n=250. How does the sample size affect the relative frequency(e.g.) counterTooLow/N of missing the mu?Explanation / Answer
The complete R program is given below:
N <- 1e5
# (a)
n <- 20
gamma_samples <- matrix(0,nrow=N,ncol=n)
for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2)
LL <- qt(0.025,n-1)
UL <- qt(0.975,n-1)
# (b)
sim_mean <- rowMeans(gamma_samples)
sum(sim_mean<LL)
sum(sim_mean>UL)
sum(sim_mean<LL)/N
# (c)
n <- 30
gamma_samples <- matrix(0,nrow=N,ncol=n)
for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2)
LL <- qt(0.025,n-1)
UL <- qt(0.975,n-1)
# (b)
sim_mean <- rowMeans(gamma_samples)
sum(sim_mean<LL)
sum(sim_mean>UL)
sum(sim_mean<LL)/N
n <- 60
gamma_samples <- matrix(0,nrow=N,ncol=n)
for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2)
LL <- qt(0.025,n-1)
UL <- qt(0.975,n-1)
# (b)
sim_mean <- rowMeans(gamma_samples)
sum(sim_mean<LL)
sum(sim_mean>UL)
sum(sim_mean<LL)/N
n <- 100
gamma_samples <- matrix(0,nrow=N,ncol=n)
for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2)
LL <- qt(0.025,n-1)
UL <- qt(0.975,n-1)
# (b)
sim_mean <- rowMeans(gamma_samples)
sum(sim_mean<LL)
sum(sim_mean>UL)
sum(sim_mean<LL)/N
n <- 250
gamma_samples <- matrix(0,nrow=N,ncol=n)
for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2)
LL <- qt(0.025,n-1)
UL <- qt(0.975,n-1)
# (b)
sim_mean <- rowMeans(gamma_samples)
sum(sim_mean<LL)
sum(sim_mean>UL)
sum(sim_mean<LL)/N
The output of a typical session is now given below:
>
> N <- 1e5 > # (a) > n <- 20 > gamma_samples <- matrix(0,nrow=N,ncol=n) > for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2) > LL <- qt(0.025,n-1) > UL <- qt(0.975,n-1) > # (b) > sim_mean <- rowMeans(gamma_samples) > sum(sim_mean<LL) [1] 0 > sum(sim_mean>UL) [1] 95413 > sum(sim_mean<LL)/N [1] 0 > > # (c) > n <- 30 > gamma_samples <- matrix(0,nrow=N,ncol=n) > for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2) > LL <- qt(0.025,n-1) > UL <- qt(0.975,n-1) > # (b) > sim_mean <- rowMeans(gamma_samples) > sum(sim_mean<LL) [1] 0 > sum(sim_mean>UL) [1] 99127 > sum(sim_mean<LL)/N [1] 0 > > n <- 60 > gamma_samples <- matrix(0,nrow=N,ncol=n) > for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2) > LL <- qt(0.025,n-1) > UL <- qt(0.975,n-1) > # (b) > sim_mean <- rowMeans(gamma_samples) > sum(sim_mean<LL) [1] 0 > sum(sim_mean>UL) [1] 99990 > sum(sim_mean<LL)/N [1] 0 > > > n <- 100 > gamma_samples <- matrix(0,nrow=N,ncol=n) > for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2) > LL <- qt(0.025,n-1) > UL <- qt(0.975,n-1) > # (b) > sim_mean <- rowMeans(gamma_samples) > sum(sim_mean<LL) [1] 0 > sum(sim_mean>UL) [1] 100000 > sum(sim_mean<LL)/N [1] 0 > > n <- 250 > gamma_samples <- matrix(0,nrow=N,ncol=n) > for(i in 1:N) gamma_samples[i,] <- rgamma(n,5,2) > LL <- qt(0.025,n-1) > UL <- qt(0.975,n-1) > # (b) > sim_mean <- rowMeans(gamma_samples) > sum(sim_mean<LL) [1] 0 > sum(sim_mean>UL) [1] 100000 > sum(sim_mean<LL)/N [1] 0
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