Question

*related to R programming*

Using simulation, estimate the coverage for the t-based interval ☺ ta/2,n-1.sx/vn for a population mean y using a confidence level of c = 0.9 with: a. a random sample of size 10 from a standard Gaussian population b. a random sample of size 10 from a standard Exponential population Use prop.test() to determine a confidence interval for the true coverage in both cases. Does the t-based interval attain the desired coverage in the Gaussian case? In the Exponential case?

Answer 1

The R-code is

N=10000# number of replicates

X1=0 #for number of samples contains CI for gausian

X2=0#for number of samples contains CI for exponential

for( i in 1:N)

{set.seed(i+i)

X=rnorm(10) #genrate random numbers from gausian distribution with mean 0 and variance 1

LL=mean(X)-qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

UL=mean(X)+qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

if(LL<0 && 0<UL) X1=X1+1

}

for( i in 1:N)

{set.seed(i+i)

X=rexp(10,1)#generate random numbers with exponential distribution with mean 1

LL=mean(X)-qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

UL=mean(X)+qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

if(LL<1 && 1<UL) X2=X1+1

}

prop.test(X1,N,0.90,conf.level=0.90)#for Gaussian

prop.test(X2,N,0.90,conf.level=0.90)#for exponential

The output of code is

>N=10000# number of replicates

>X1=0 #for number of samples contains CI for gausian

>X2=0#for number of samples contains CI for exponential

>for( i in 1:N)

>{set.seed(i+i)

>X=rnorm(10) #genrate random numbers from gausian distribution with mean 0 and variance 1

>LL=mean(X)-qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

>UL=mean(X)+qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

>if(LL<0 && 0<UL) X1=X1+1

>}

>for( i in 1:N)

>{set.seed(i+i)

>X=rexp(10,1)#generate random numbers with exponential distribution with mean 1

>LL=mean(X)-qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

>UL=mean(X)+qt(1-0.05,9)*(sd(X)/sqrt(length(X)))

>if(LL<1 && 1<UL) X2=X1+1

>}

>prop.test(X1,N,0.90,conf.level=0.90)#for Gaussian

-sample proportions test with continuity correction

data: X1 out of N, null probability 0.9

X-squared = 1.0336, df = 1, p-value = 0.3093

alternative hypothesis: true p is not equal to 0.9

90 percent confidence interval:

0.8980735 0.9079062

sample estimates:

p

0.9031

>prop.test(X2,N,0.90,conf.level=0.90)#for exponential

1-sample proportions test with continuity correction

data: X2 out of N, null probability 0.9

X-squared = 1.1025, df = 1, p-value = 0.2937

alternative hypothesis: true p is not equal to 0.9

90 percent confidence interval:

0.8981757 0.9080040

sample estimates:

p

0.9032

# The CI for desired coverage in case of standard gausian distribution is(0.8980735 ,0.9079062) which contains true coverage

# The CI for desired coverage in case of standard exponential distribution( 0.8981757, 0.9080040) which contains true coverage