Question

Using Rcode solve

A company with a large fleet of cars wants to study the gasoline usage. They check the gasoline usage for 50 company trips chosen at random, finding a mean of 25.02 mpg and sample standard deviation is 4.83 mpg.

a. Which kind of confidence interval is appropriate to use here, z-interval or t-interval?

b. What are the assumptions to check for the interval you chose?

c. Please use R to find the critical value the company needs when constructing a (two-sided) 98% CI.

d. Please use R to construct a (two-sided) 98% CI for the mean of the general gasoline usage.

e. Please use R to construct a 98% upper confidence bound for the mean of the general gasoline usage.

f. Create a R function whose argument is the width of CI, and the output is the sample size necessary to achieve such accuracy. The confidence level is fixed at 98%.

g. Apply the function you created in part (f) to demonstrate that larger sample size is required to achieve better accuracy (i.e, narrower CI width). Confidence level is fixed at 98%. Show at least three examples

Answer 1

Given,

a) We will use z- interval, why, is in the next part.

b) Since, sample size is large (n>30), so, we could preferably use z- interval

Assumptions are:

c).

Confidence level = 98%

Critical value for two sided test is,

qnorm(1-0.02/2) ### this will give

2.326348

For two sided test we have

d).

To construct 98% confidence interval, we develop a function in R,

confint=function(xbar,s,n){
xbar=25.02
s=4.83
n=50
moe=2.33*s/sqrt(n) ### moe is margin of error
Llimit=xbar-moe
Ulimit=xbar+moe
int=c(Llimit,Ulimit)
return(int)
}
confint(xbar,s,n)
23.42846 26.61154

Hence, the inerval is (23.42846 , 26.61154)

e)

For upper bound mean for right tailed test (or one sided test),

confint=function(xbar,s,n){
xbar=25.02
s=4.83
n=50
moe=2.05*s/sqrt(n) ### moe is margin of error
Llimit=xbar-moe
Ulimit=xbar+moe
int=c(Llimit,Ulimit)
return(int)
}
confint(xbar,s,n)
23.61972 26.42028

Hence, the interval is (23.61972 , 26.42028)

• The first assumption is that our sample is a simple random sample and this information is usually provided in the problem.

• Our sampling distribution is normally distributed