Applied Statistics-Homework 3 Solved

Exercise 1: Datasaurus
For this question, we’ll use the data contained in the datasaurus_dozen dataset within the datasauRus package. Make sure that you have installed and loaded the datasauRus package before you begin working on this exercise.

The datasaurus_dozen contains three variables:

•            dataset, with 13 options

•            x, and

•            y.

It may help to look at the first few rows of the datasaurus_dozen dataset.

head(datasaurus_dozen)

## # A tibble: 6 x 3

##        dataset            x           y

## 1 dino
55.4 97.2
## 2 dino
51.5 96.0
## 3 dino
46.2 94.5
## 4 dino
42.8 91.4
## 5 dino
40.8 88.3
## 6 dino
38.7 84.9
## <chr> <dbl> <dbl> part a

Let’s begin by creating the following four datasets:

•            Create a dino object in R for those observations in the datasaurus_dozen dataset that take the value dino for the variable dataset.

•            Create a high_lines object in R for those observations in the datasaurus_dozen dataset that take the value high_lines for the variable dataset

•            Create a star object in R for those observations in the datasaurus_dozen dataset that take the value star for the variable dataset

•            Create a x_shape object in R for those observations in the datasaurus_dozen dataset that take the value x_shape for the variable dataset.

# Use this code chunk to answer this question. dino = subset(datasaurus_dozen, datasaurus_dozen$dataset == 'dino') high_lines = subset(datasaurus_dozen, datasaurus_dozen$dataset == 'high_lines')

star = subset(datasaurus_dozen, datasaurus_dozen$dataset == 'star') x_shape = subset(datasaurus_dozen, datasaurus_dozen$dataset == 'x_shape')

part b
For each of the four R objects you created in part a, report the following statistics:

•      number of rows & columns

•      mean of x

•      mean of y

# Use this code chunk to answer this question. dim(dino)

## [1] 142           3

dim(high_lines)

## [1] 142           3

dim(star)

## [1] 142           3

dim(x_shape)

## [1] 142           3

mean(dino$x)

## [1] 54.26327

mean(high_lines$x)

## [1] 54.26881

mean(star$x)

## [1] 54.26734

mean(x_shape$x)

## [1] 54.26015

mean(dino$y)

## [1] 47.83225

mean(high_lines$y)

## [1] 47.83545

mean(star$y)

## [1] 47.83955

mean(x_shape$y)

## [1] 47.83972 part c

For each of these four R objects, report the following:

•     the correlation of x and y

•     the coefficients for the linear model predicting y from x

# Use this code chunk to answer this question. cor(dino$x, dino$y)

## [1] -0.06447185

cor(high_lines$x, high_lines$y)

## [1] -0.06850422

cor(star$x, star$y)

## [1] -0.0629611

cor(x_shape$x, x_shape$y)

## [1] -0.06558334

dino_lm = lm(y ~ x, data = dino) high_lines_lm = lm(y ~ x, data = high_lines) star_lm = lm(y ~ x, data = star) x_shape_lm = lm(y ~ x, data = x_shape) coef(dino_lm)

## (Intercept)                               x

## 53.4529784 -0.1035825

coef(high_lines_lm)

## (Intercept)                               x

## 53.8087932 -0.1100695

coef(star_lm)

## (Intercept)                               x

##        53.326679         -0.101113

coef(x_shape_lm)

## (Intercept)                               x

## 53.5542263 -0.1053169 part d

What do you notice from your results in parts b & c? What might be the underlying cause of your results from each of these datasets?

Note: there is a correct answer for the first question of d. The second question is asking you to speculate as to what might be occurring without a correct answer.

part e
Graph each of the datasets, using the x and y variables for their respective axes. Include the dataset name in the title of each graph. Axes labels of x and y are sufficient for this problem.

# Use this code chunk to answer this question.

ggplot(data = dino, aes(x = x, y = y)) + geom_point() +

labs(x = 'X Values', y = 'Y Values', title = 'Dino Dataset')
Dino Dataset
 

ggplot(data = high_lines, aes(x = x, y = y)) + geom_point() +

labs(x = 'X Values', y = 'Y Values', title = 'High Lines Dataset')
High Lines Dataset
 

ggplot(data = star, aes(x = x, y = y)) + geom_point() +

labs(x = 'X Values', y = 'Y Values', title = 'Star Dataset')
Star Dataset
 

ggplot(data = x_shape, aes(x = x, y = y)) + geom_point() +

labs(x = 'X Values', y = 'Y Values', title = 'X Shape Dataset')
X Shape Dataset
 

part f
What did you observe from the graphs? What is a takeaway point from this exercise?

 

Exercise 2: Variable Types
Indicate the variable type for each of the variables described. Be sure to be specific, including the general and specific labels (e.g. quantitative discrete).

part a
Popularity of a roller coaster, as measured by the number of riders on September 1, 2022. Answer: quantitative discrete

part b
Theme park where a roller coaster is located. Answer: nominal categorical

part c
Time (in minutes) a roller coaster was operating on September 1, 2022.

part d
Target audience for the roller coaster, based on age grouping.

part e
How many breakdown events resulting in a pause from operation that a roller coaster experienced between September 1, 2020 and August 31, 2021.

 

Exercise 3: Variable Roles & Study Types
For each of the following proposed studies, indicate the variable roles for each variable described. Is the study described experimental or observational?

part a
Yanis suspects that the eldest child in a family grows to be the shortest adult, and that the youngest child grows to be tallest. Berza reminds Yanis that adult height is also affected by sex, so Yanis decides to record that as well for all participants. Yanis recruits adult participants with at least one sibling for this study.

Variables:

•      Birth order (eldest vs. youngest)

•      Adult Height

•      Sex

part b
Do your friends approach mealtime the same way that you do? Some students report when they come to college that they eat faster than their friends do. One student, Alex, speculates that the speed with which you eat a meal is determined by where you are from geographically. Jennifer reminds him that additional factors, like how much you talk while you eat, are also related both to your geographic region and to how long it takes to eat a meal. Fernando finds this theory interesting, and so decides to gather data on these variables from a campus dining hall.

Variables:

•      Meal Time

•      Geographic Region

•      Talk Time during Meal

part c
Inspired by Fernando’s study in the dining hall, Brenda designs her own study. Brenda wants to know if

ordering choices and eating time depend on how many people you are seated with. Brenda designs a study where entrants to the dining hall are randomly assigned to eat at a table by themselves, with 1 friend, with 2 friends, or with 3 friends. The food ordered and the time spent eating are both recorded.

Variables:

•      Ordering Choices

•      Meal Time

•      Number of Dining Companions

 

Exercise 4: Birthweight, Descriptive Summaries
In this exercise, we’ll work with the birthwt dataset contained within the MASS package. Read through the documentation using the Help command below. If you would like to prevent new browser windows from reopening every time you knit the document, you may opt to comment this line of code out by adding a hashtag at the beginning of the line.

 

part a
How many observations are in this dataset (use R function)? How many variables (use R function)? Where and when was this data collected (not from an R function)? Provide these details in a sentence after your code block.

# Use this code chunk for your answer. dim(birthwt)

## [1] 189 10

part b
We’ll be using this dataset to predict baby’s birthweight using the other variables in the dataset. In this part, we’ll think about reasons that causality might be plausible for this specific scenario and reasons causality might not be supported based on the underlying behavior of the variables of interest. Without performing any numerical analyses, what reason(s) support a causal relationship between the other variables, excluding the indicator of a low birth weight, and the baby’s birthweight? What reason(s) undermine any determination of causality or suggest that causality might not be a reasonable explanation for these variables?

part c
Create a correlation matrix of the baby’s birthweight, the mother’s age, and the mother’s weight. Which of the possible explanatory variables has the highest correlation with the baby’s birthweight?

# Use this code chunk for your answer. cor(birthwt[,c(10, 2, 3)])

##           bwt              age              lwt ## bwt 1.00000000 0.09031781 0.1857333 ## age 0.09031781 1.00000000 0.1800732 ## lwt 0.18573328 0.18007315 1.0000000

part d
A doctor would like to analyze the birth weight data with an aim of generating results that could be applied to her current patients. Is this an appropriate use of the dataset? Explain.

part e
Thinking critically about this data, are there additional variables that could be added? Do you have concerns about how this data might be used? Any additional information you’d like to know about the data?

 

Exercise 5: Interpreting a Linear Model for Birthweight [25 points]
We’ll continue analyzing the birthwt dataset that we started looking at in the last Exercise. For this question, we’ll focus on the variables bwt and lwt.

part a
Visualize the relationship between the mother’s pre-pregnancy weight and the baby’s weight. Make sure to also provide appropriate titles and axes labels for your graph. Then, interpret this relationship.

# Use this code chunk for your answer. ggplot(data = birthwt, aes(x = lwt, y = bwt)) + geom_point() +

labs(title = "Mother's Weight vs Child's Birth Weight", x = "Mother's Birth Weight", y = "Child's Birth Weight")
Mother's Weight vs Child's Birth Weight
 

part b
Fit a linear model that predicts the baby’s weight from the mother’s pre-pregnancy weight. Write that model out below.

# Use this code chunk for your answer. lm = lm(bwt ~ lwt, data = birthwt) summary(lm)

##

## Call:

## lm(formula = bwt ~ lwt, data = birthwt)

##

## Residuals:

##           Min        1Q              Median  3Q              Max ## -2192.12 -497.97   -3.84      508.32 2075.60

##

## Coefficients:

##                                      Estimate Std. Error t value Pr(>|t|)

## (Intercept) 2369.624                      228.493 10.371           <2e-16 ***

## lwt                             4.429              1.713        2.585        0.0105 *

## ---

## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

##

## Residual standard error: 718.4 on 187 degrees of freedom

## Multiple R-squared: 0.0345, Adjusted R-squared: 0.02933 ## F-statistic: 6.681 on 1 and 187 DF, p-value: 0.0105

part c
Interpret each of the fitted coefficients (intercept and slope) for this model.

part d
Calculate the estimated mean baby’s birthweight for a mother with a pre-pregnancy weight of 147 pounds. What is the residual for a mother with a pre-pregnancy weight of 147 and a baby’s birthweight of 3000 g.

# Use this code chunk (if needed) for your answer.

3000 - (2369.624 + (4.429 * 147))

## [1] -20.687

part e
One of the mother’s weights was accidentally removed from the dataset. However, we know the corresponding baby’s weight (2743 g) and the residual (-40 g). What was the original mother’s weight?

# You may use this code chunk for your calculation, or you may type your calculation below. observed = 2743 # -40 = 2743 - x

predicted = (-2743 - 40)/-1 predicted
## [1] 2783

# predicted = 2369.624 + (4.429 * x)

(2783 - 2369.624)/4.429

## [1] 93.33394

# check answer

2369.624 + (93.33394* 4.429 )

## [1] 2783

observed - (2783)

## [1] -40

 

Exercise 6: Formatting
The last five points of the assignment will be earned for properly formatting your final document. Check that you have:

•      included your name on the document

•      properly assigned pages to exercises on Gradescope

•      select page 1 (with your name) and this page for this exercise (Exercise 6)

•      all code is printed and readable for each question

•      generated a pdf file