PSTAT 100: ICA 02 Review

Some Extra Questions

Ethan P. Marzban

Department of Statistics and Applied Probability; UCSB

Summer Session A, 2025

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Division of Labor

Within each of your groups (tables), assign one person to be:

The scribe (who will write your group’s answers on the board)
The timekeeper (the one who ensures your group is staying on-task)

As the ICA is closed-book, please put away all electronics and use only your note sheets.

Question 1: Squirrel!

Setup, and Part (a)

Two common types of squirrels in California are the Tree Squirrel and the Ground Squirrel. These species differ in several characteristics, one of which being weight. To that end, we consider building a classifier to classify a squirrel as either "Tree" or "Ground" based on its weight (in lbs).

Part (a):

Encoding "Tree" as a value of 1, the following regression table is outputted:

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  2.24749    0.28725   7.824 2.42e-12 ***
wt          -0.12093    0.01897  -6.376 3.68e-09 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

What is the probability that a 12-pound squirrel is a Tree Squirrel? Round to four decimal places.

03:30

Question 1: Squirrel!

Part (b)

Part (b)

We can use the fitted model from part (a) to generate a ROC curve, pictured below with some corresponding threshold values:

Based on the ROC Curve to the left, what is the ideal threshold for building a classifier, and how can you tell?

03:00

Question 1: Squirrel!

Part (c)

For Parts (c) - (d): We now consider the “opposite” problem: that is, we wish to predict a squirrel’s weight given its species. Our data is stored in a data frame called squirrels, containing two columns: species (the species, either "Ground" or "Tree"), and wt (the weight, in lbs).

Part (c)

Consider the model: \(y_i = \beta_0 + \beta_1 1 \! \! 1\{\texttt{species}_i = \texttt{Tree}\} + \varepsilon_i\) where we impose the “usual” assumptions on \(\varepsilon_i\), and \(y_i\) denotes the i^th squirrel’s weight.

What are the OLS estimates for β₀ and β₁? For your convenience, we have provided the following summaries:

species	Avg. Weight
Ground	15.92
Tree	13.8

03:00

Question 1: Squirrel!

Part (c)

Part (d)

Here is some R code and its output:

squirrels %>% pull(species) %>% levels()

[1] "Ground" "Tree"

Based on this, suppose we ran lm(species ~ wt, data = "squirrels"). Which species would R use as the baseline series? Based on your answer to this question, is this call to lm() fitting the same model as in Part (c) or not? Explain.

03:00

Question 2: We All Scream for Ice Cream

GauchoCream is a new brand of ice cream using a sweetener that it claims “tastes just like sugar.” To test this, Morgana recruits 100 UCSB student volunteers and randomly divides them into two groups of 50 each. To one group, she gives a scoop of GauchoCream ice cream and to the other group she gives a scoop of Thrifty ice cream (which is sweetened with sugar).

She asks each of the 100 participants whether they believed their ice cream was sweetened with sugar, or an artificial sweetener, and records these observations. She compares the proportion of each group that correctly identified their ice cream’s sweetener and finds: 70% of those who had Thrifty ice cream correctly identified it as being sweetened with sugar, but only 20% of those who had GauchoCream ice cream correctly identified it as being artificially sweetened.

Is this an Experiment or an Observational Study? Is it Longitudinal or Cross-Sectional? Explain.

04:00

Question 3: Density

We have taken a sample of size one from an unknown distribution; this sampled value is 1. Sketch the Kernel Density Estimate (KDE) of the true density f_X(x), using a boxcar kernel with a bandwidth of 1. Clearly indicate your axis labels and values!

03:00

Question 4: Regression

Setup

We fit a regression of a variable y onto three covariates: x1, x2, and x3. Some output is provided below:

Output 1
Output 2
Output 3

lm(y ~ ., data = df1) %>% summary()


Call:
lm(formula = y ~ ., data = df1)

Residuals:
     Min       1Q   Median       3Q      Max 
-2.98565 -0.69137  0.07014  0.74795  2.27946 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)  0.56932    0.21328   2.669  0.00864 ** 
x1           1.16899    0.03971  29.441  < 2e-16 ***
x2           1.77788    0.05016  35.447  < 2e-16 ***
x3           2.09877    0.01000 209.831  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 1.044 on 121 degrees of freedom
Multiple R-squared:  0.9976,    Adjusted R-squared:  0.9976 
F-statistic: 1.712e+04 on 3 and 121 DF,  p-value: < 2.2e-16

cor(df1)

           x1        x2         x3         y
x1 1.00000000 0.6889297 0.02421951 0.3500102
x2 0.68892972 1.0000000 0.00478940 0.3433797
x3 0.02421951 0.0047894 1.00000000 0.9305597
y  0.35001022 0.3433797 0.93055970 1.0000000

summary(df1)

       x1               x2               x3                 y         
 Min.   :-4.923   Min.   :-1.416   Min.   :-19.0874   Min.   :-36.85  
 1st Qu.: 2.378   1st Qu.: 3.582   1st Qu.:  0.4951   1st Qu.: 13.22  
 Median : 3.993   Median : 4.955   Median :  6.8132   Median : 27.30  
 Mean   : 4.040   Mean   : 5.049   Mean   :  5.4186   Mean   : 25.64  
 3rd Qu.: 6.366   3rd Qu.: 6.760   3rd Qu.: 11.6295   3rd Qu.: 41.01  
 Max.   :13.127   Max.   :13.785   Max.   : 34.7374   Max.   : 85.44

Question 4: Regression

Questions

Part (a): Holding all else constant, to what average increase in y does a one-unit increase in x2 correspond?

Part (b): A new observation has (x1, x2, x3) values (25, 0, -13). Either use the model to predict the associated response variable, or explain why doing so is not a good idea.

Part (c): Is there evidence of multicollinearity in the model? How can you tell?

Part (d): What are the two main assumptions we make on the noise in our model, and how are they reflected in a residuals plot?

05:00