Data Analysis 5 - Simulations and Confidence Intervals

Part 1. (12 Points)

Each year the EPA does an analysis on the current models of vehicles sold the United States. The data provided in the data set EpaFE2019Data.csv is a subset of this analysis, if you are curious you may access the full data set from the EPA website http://www.fueleconomy.gov/feg/download.shtml.

Use the R script titled DA5_Simulations_and_Confidence_Intervals.R to upload the EpaFE2019Data.csv dataset and perform the necessary simulations.

In this exercise, we will use EPA car data as an example of a population.

• We will use R to select a simple random sample of vehicles from the population.
• We will then use this sampled data to compute confidence intervals and perform hypothesis tests.
  • This means, unlike typical hypothesis test or estimation procedures, we know our population parameters.
• Why should we do this?
  • To provide an opportunity to evaluate the validity of estimation and hypothesis testing procedures. Does it work like we say it should?

Follow the comments in the R script to complete the following: The variable combined carbon dioxide emissions, or CombCO2, represents the combined city and highway carbon dioxide emissions for vehicles sold in the US.

1. (3 points) Make a histogram of this variable and include your histogram here. What are the values of ? How large is the population? Note: Consider this data a population. This implies the mean and standard deviation are parameters. Paste the histogram and give a brief description of the population data.

Pop. Size: 896
μ/Mean: 399.8717
σ/SD: 89.82545
The distribution is fairly symmetric with some slight outliers to the right.

2. (3 points) Take a random sample of size 45 from the population. From your sample, calculate the sample statistics, and s. Make a histogram of carbon dioxide emissions for the sample of 45 vehicles. Paste the histogram. Make a brief description of the sampled data. Does it look much like the population?

X-bar: 398.9778
SD: 89.52285
The data is skewed right with a median of 375 and a spread of 350.

3. (3 points) Use , your sampled mean, from (Part 1b) and your sample standard deviation (Part 1b), to calculate the 90% confidence interval (CI) for . Show work!

398.9778 +- 1.645(89.52285/451/2) -> [377, 421]

4. (1 point) Does your interval from part c include the true population mean for fuel efficiency?

Yes

5. (2 points) There are 400 students this term completing this same assignment. Assuming they calculated the CI correctly, how many students should we expect to have an interval that does not contain the true mean?

With a 90% confidence interval, 10% should not contain the true mean.

Part 2. (11 Points)

A newborn is considered to be premature if it is born before 37 weeks of its gestation period. Assume the proportion of all babies in U.S. that are born prematurely is 0.101. This is the population parameter. We’ll use this parameter to simulate many samples in and make observations about the behavior of confidence interval intervals for a proportion.

To answer the following questions, you’ll need to run the R code given in DA5_Simulations_and_Confidence_Intervals.R in the section labeled Part 2.

1. (1 point) Randomly generate a single sample proportion p^, for a sample of size n = 40 from the population described above. Report your value for p^ here. Use lines 54-69 in the R script to do this.

p^ = 0.175

2. (2 points) Use p^ from the previous question to construct a 99% confidence interval for the proportion newborns born prematurely. Show your work and report your confidence interval here.

0.175 +- 2.58(0.175(1 - 0.175)/40)1/2
0.175 +- 2.58((0.175*0.825)/40)1/2
0.175 +- 2.58(0.144375/40)1/2
0.175 +- 2.58(0.003609375)1/2
0.175 +- 2.58(0.0600780742035)
0.175 +- 0.155001431445
= [0.02 ,0.33]

3. (1 point) Does the interval you constructed in the previous question contain the population proportion p = 0.101?

Yes

4. (1 point) If you were to collect 10,000 random samples, all of size n=40, and construct a 99% confidence interval for the proportion, what percentage of those intervals would you expect to capture the true population proportion p = 0.101?

99%

5. (2 points) Using R, simulate 10,000 random samples and construct the 99% confidence interval for each sample. I’ve written the code for you to do this. I have provided the code and some comments to explain what the code is doing in the DA5_Simulations_and_Confidence_Intervals.R (lines 71-93); however, if you still have questions about the code or want to learn about R syntax, just ask! Report the percentage of intervals that successfully captured the population proportion. Is this value the same or close to the value you expected (your answer to Part 2d)?

After 10,000 random samples, 92.74% of the 99% confidence intervals contained
the true mean. This is somewhat close to my expectation, however still lower
than I thought it would be.

6. (2 points) Before constructing the confidence interval in Part 2b, you were not instructed to check the conditions necessary to construct a confidence interval for the population proportion. In order for the confidence intervals to behave as expected, we need the following two conditions to be met: n >= 10 and n(1 - p) >= 10. Show that these conditions are not met.

40(.101) >= 10 & 40(1 - .101) >= 10
4.04 >=10 & 35.96 >= 10
4.04 < 10 -> np < 10

7. (2 points) Using your answers from parts d and e, comment on how the confidence intervals behave when the sample size condition is not met.

When the sample size condition is not met, the confidence interval is inaccurate
and may not actually be a 99% confidence interval.

Gradescope Page Matching (2 points)

When you upload your PDF file to Gradescope, you will need to match each question on this assignment to the correct pages. Video instructions for doing this are available in the Start Here module on Canvas on the page “Submitting Assignments in Gradescope”. Failure to follow these instructions will result in a 2-point deduction on your assignment grade. Match this page to outline item “Gradescope Page Matching”.