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

SolutionA:

since population standard deviation is not gievn use Z interval.

since sample standard deviation is given use t interval

SolutionB:

distribution to be normal

sample is simple random sample

here n=50

n>30

according to centarl limit theorem follows normal distribution

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

n <- 50
criticalt <- qt(0.98,df=n-1)

2.109873

CRITICAL T =2.109873

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

Rcode:

xbar <- 25.02
s <- 4.83
n <- 50
margn_of_error <-qt(0.98,df=n-1)*s/sqrt(n)
lowerlimit <- xbar-margn_of_error
lowerlimit
upperlimit <- xbar+margn_of_error
upperlimit

lowerlimit
23.57882

upperlimit:

26.46118

98% confidence interval for true mean lies in between 23.57882 and 26.46118

Solutione:

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

upperlimit <- xbar+margn_of_error
upperlimit

26.46118