Applied Statistics-Homework 1 Solved

Exercise 1: Initial Setup
The first five points for this assignment come from setting up the document and your environment correctly. You will earn these points by knitting your document and creating a pdf.

I recommend that you knit the document to a pdf now, before beginning the main coding exercises. This way, you ensure that your environment is set up correctly and know that the document was properly formatted before you begin editing the document. To knit the document, click the ball of yarn with a knitting needle on it or the word “Knit” beside it.

Exercise 2: Formatting & Submitting
The next 5 points of this assignment will be earned for completing the following tasks:

•      Including your name in the header of the document

•      Assigning pages correctly on Gradescope during submission

Please also assign page 1 with your name for this exercise.

 

Exercise 3: Calculations
R is a powerful calculator. Let’s perform some basic operations with R in this exercise.

part a
Translate the following mathematical statement into R code, and calculate the result:

(7 + 5)9−3× 

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

((7 + 5)ˆ6) * (55/6)

## [1] 27371520

part b
In addition to using R as a calculator, we can use built-in functions of R to simplify our calculations. Below, use the mathematical constant π ≈ 3.14159, represented by pi in R, to calculate the value of π2.3. Make sure this value is printed.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

piˆ(2.3)

## [1] 13.91377

part c
Create a vector that consists of the integers from 3 to 9, including both 3 and 9. Assign the vector to the variable z.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

z <- (3:9) z
## [1] 3 4 5 6 7 8 9

part d
Using the vector z that you created in part c, generate a new vector y with values of πz − π2.3. Print y.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

y <- (piˆz) - piˆ(2.3) y
## [1]             17.09251          83.49532        292.10592           947.47543 3006.37946 9474.61725

## [7] 29795.18557

part e
We can chain functions and operations, allowing us to use multiple operations and functions in one line. In one line of code, square the values contained in each entry of y, then sum all of those values, and finally take the square root of the resulting sum. Print the resulting value.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

print(sqrt(sum(yˆ2)))

## [1] 31425.31

 

Exercise 4: Dataset Basics
While it can be very helpful that R is powerful in an abstract situation, we often want to answer questions using data. The remaining exercises will allow us to apply some of the built-in statistical functions to a dataset. We’ll look at a road casualties dataset from Great Britain in the 1960s to 1980s. You can read more about the dataset and variables using the Help feature to the right or by typing ?Seatbelts into the R Console below.

sb = as.data.frame(Seatbelts) sb$law = as.factor(sb$law)

The dataset is now stored in R as sb.

Hint: the Day 1 Demo file on Canvas will be helpful for Exercises 4-7.

part a
Using R code, determine how many columns (variables) and rows (months) are contained in the dataset. Write the solution below the code, replacing the blank lines with the appropriate numbers.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

dim(sb)

## [1] 192            8

 

part b
Print the first 6 rows of the dataset.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it. head(sb)

##                DriversKilled drivers front rear                   kms PetrolPrice VanKilled law

## 1       107        1687      867 269 9059              0.1029718            12          0 ## 2              97          1508      825 265 7685              0.1023630            6            0 ## 3              102        1507      806 319 9963              0.1020625            12          0 ## 4              87          1385      814 407 10955              0.1008733            8            0 ## 5              119        1632      991 454 11823              0.1010197            10          0

## 6                         106         1511         945 427 12391          0.1005812                 13                                  0

part c
Looking at the first 6 rows of the dataset that you’ve printed above, what do you notice? What questions do you have about this dataset?

Exercise 5: Numerical Summaries
Suppose that the Department of Transportation is interested in comparing the number of drivers killed or seriously injured (drivers) with the number of front-seat passengers killed or seriously injured (front).

part a
First, generate numerical summaries for the drivers variable. Be sure to calculate the mean, the five number summary, and the standard deviation (try searching on Google or looking ahead in the notes for this function if you haven’t seen it in class yet).

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

summary(sb$drivers)

##          Min. 1st Qu. Median              Mean 3rd Qu.  Max. ##              1057              1462              1631              1670              1851              2654

sd(sb$drivers)

## [1] 289.611

part b
For the drivers variable: Which is larger, the mean or the median? Think back to what this might tell you about the shape of the distribution. What do you anticipate about the shape of the distribution from the mean, the median, and the five number summary?

Note: We won’t grade your prediction based on correctness; we’re hoping that you’ll think about what the data means here.

.

part c
Now, calculate the same numerical summaries from part a for the front variable.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

summary(sb$front)

## Min. 1st Qu. Median Mean 3rd Qu. Max. ## 426.0 715.5 828.5 837.2 950.8 1299.0

sd(sb$front)

## [1] 175.099

part d
Compare the numerical summaries for the drivers from part a with the numerical summaries for the front seat passengers in part c. What do you notice? What real world implications might this have?

 

Exercise 6: Visualizing and Interpreting One Variable
We generated numerical summaries for the number of deaths and serious injuries of drivers and of front seat passengers in the last problem. Now, let’s visualize what the distributions for these variables look like.

To do this, we’ll load the ggplot2 package, which allows you to create graphics like we did in class.

library(ggplot2)

part a
First, generate a histogram of the drivers variable.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

ggplot(data = sb, aes(x = drivers)) + geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

 

part b
Now, generate a histogram of the front variable.

# Use this code chunk to answer the question, by replacing this line or # adding a new line below it.

ggplot(data = sb, aes(x = front)) + geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

 

part c
How would you describe the number of deaths and serious injuries of drivers and of front seat passengers to an intern in the Department of Transportation? What do you notice from these two graphs?

 

Exercise 7: Scatterplots of Two or More Variables
Finally, we’ll create a scatterplot of the relationship between deaths and serious injuries of drivers compared to those of front seat passengers.

To do this, we’ll load the ggplot2 package, which allows you to create graphics like we did in class. You should delete the hashtag and space at the beginning of this line of code to load the package. Note that this line of code will not run if you have not previously installed the ggplot2 package as described at the beginning of this Homework1.Rmd file.

library(ggplot2)

part a
Modify the dataset name and variable names from the following code found in the Day 1 Demo R file to create a scatterplot of the Seatbelts data. Place the deaths and serious injuries from drivers on the x axis and the deaths and serious injuries from front seat passengers on the y axis.

ggplot(data = coasters, mapping = aes(y = Speed, x = Height)) + geom_point()

# Use this code chunk to answer the question by replacing this line or # adding a new line below it.

ggplot(data = sb, aes(x = drivers, y = front)) + geom_point() +

geom_smooth(method = 'lm', se = F)
## `geom_smooth()` using formula 'y ~ x'

 

part b
Now, add in the variable law, which represents whether a law requiring the use of seatbelts was in place for the given month or not. Incorporate the law variable in the shape and color of the points in the scatterplot. Add lines of best fit to this graph, as well. Below, you will find sample code from the Day 1 Demo R file.

ggplot(data = coasters, mapping = aes(y = Speed, x = Height, shape = Track, color = Track)) + geom_point()

+ geom_smooth(method = ‘lm’, se = F)

# Use this code chunk to answer the question by replacing this line or # adding a new line below it.

ggplot(data = sb, aes(x = drivers, y = front, shape = law, color = law)) + geom_point() +

geom_smooth(method = 'lm', se = F)
## `geom_smooth()` using formula 'y ~ x'

 

part c
What do you notice from the scatterplots generated in parts a & b above?

part d
What would be the next step that you would want to take if you were to continue looking at this dataset? You may choose to answer some or all of the following prompts for this question.

•      If you were to continue analyzing the drivers and/or front variables, what would you take as the next step?

•      What questions would you want to explore using any or all other variables in the dataset?

•      What other visualizations might you pursue?

•      If you could have any additional variable or information added to the dataset, what would you ask for? How would you use that information?

•      Thinking about the data more deeply, do you have any concerns about what might be missing from the data? What types of information might not have been recorded? Are there any ambiguous variables or situations in the data?

There is not one correct answer to this question; you can be creative in your approach. No need to perform any suggested analyses for this problem.

 

Exercise 8: Debugging
This coding exercise includes opportunities for debugging (fixing) common errors in RMarkdown. For this exercise, you will be provided with chunks of code that will prevent you from knitting the document, have some error in them, or are not formatted correctly. Because these chunks have errors, they also have an exclamation mark and a space (! ) added at the beginning of the lines to allow you to knit the document initially. Please remove the exclamation marks and spaces at the beginning of the lines, fix the errors, and then knit the resulting chunk. Additionally, explain the error that you corrected.

For this exercise, you do not need to change any R code or functions; focus on the formatting of the chunks, successfully knitting the document, and having appropriate formatting for the document.

Hint: Work through one part at a time to help isolate and correct any errors. Knit the document now to ensure there are no other errors present in the document prior to starting this exercise.

part a
First, remove the exclamation mark and space that are at the beginning of the next three lines of code. Then try knitting the document. You should see an error. Make adjustment(s) to the next three lines of code until the document successfully knits. Then note what you changed or what error was initially present.

sum(1:10)

## [1] 55

part b
sum(11:20)

## [1] 155

part c

(1:10)ˆ2

## [1]            1      4                 9 16 25 36 49 64 81 100

part d
library(ggplot2)

ggplot(data = pressure, mapping = aes(x = temperature, y = pressure)) + geom_point() +
geom_smooth(metho

## `geom_smooth()` using formula 'y ~ x'

 

Hint: This part results in a formatting error. It does not generate an error that prevents the document from successfully knitting.