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Statistical Analysis Exercises with R

## Exercise 1: Confidence interval for gasoline usage

Exercise 1: 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. Based on the past experience, they believe the standard deviation of the general gasoline usage is 4.83 mpg.

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

(b) Please use R to find the critical value they need when constructing a 95% CI.

(c) Please use R to construct a 95% CI for the mean of the general gasoline usage.

(d) If they want to control the width of CI to be within 0.8 mpg, at least how many trips do they have to sample? Use R to calculate.

(e) Create a R function whose argument is the width of CI, and the output is the sample size necessary to achieve such accuracy. Confidence level is fixed at 95%.

(f) Apply the function you created in part (e) to demonstrate that larger sample size is required to achieve better accuracy (i.e, narrower CI width). Confidence level is fixed at 95%.

Exercise 2: In a class survey, students are asked how many hours they sleep per night. In the sample of 22 students, the mean was 5.77 hours with a standard deviation of 1.572 hours. The parameter of interest is the mean number of hours slept per night in the population from which this sample was drawn.

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

(b) Please use R to find the critical value they need when constructing a 98% CI.

(c) Please use R to construct a 98% CI for the mean number of hours slept per night.(d) Please use R to construct a 98% lower confidence bound for the mean number of hours slept per night.

Exercise 3: In the year 2001 Youth Risk Behavior survey done by the U.S. Centers for Disease Control, 747 out of 1168 female 12th graders said they always use a seatbelt when driving. Lets construct a 98% confidence interval for the proportion of 12th grade females in the population who always use a seatbelt when driving.

(a) Use R to find the score CI for the proportion of 12th grade females in the population who always use a seatbelt when driving.

(b) Use R to find the traditional CI for the proportion of 12th grade females in the population who always use a seatbelt when driving.

(c) Assuming there is no prior information or past experience available, what is the sample size necessary to control the score 98% CI width to be within 0.01.

(d) Assuming there is no prior information or past experience available, what is the sample size necessary to control the traditional 98% CI width to be within 0.01.

Exercise 4: Consider the problem in Exercise 1 again. The company wants to conduct a test, with α = 0.05, to see whether the fleet average is less than 26 mpg.

(a) Which kind of tests is appropriate to use here, z-test or t-test?

(b) Write appropriate hypotheses. Is the alternative hypothesis one-sided or two-sided?

(c) Use R to compute the test statistic and construct rejection region. Make conclusion using the test statistic and rejection region.

(d) Use R to compute p-value. Make conclusion using p-value. Is it consistent with the conclusion in part (c)?