STA 141A: Assignment 2
Amanda Li 920562233
Instructions You may adapt the code in the course materials or any sources (e.g., the Internet, classmates, friends). In fact, you can craft solutions for almost all questions from the course materials with minor modifications. However, you need to write up your own solutions and acknowledge all sources that you have cited in the Acknowledgement section.
Failing to acknowledge any non-original efforts will be counted as plagiarism. This incidence will be reported to the Student Judicial Affairs.
library(nycflights13)- Pick a data set that you find interesting. This data set should contain at least four variables. Briefly explain the data set (background, source, variables, samples, etc.), and why it interests you.
The dataset I chose is a pre-installed dataset in R Studio called mpg. I chose this dataset because I have an interest in cars. It has 234 observations and 11 columns (variables). The variables include manufacturer, model, displ, year, cyl, trans, drv, cty, hwy, fl, class. Each observation is a car.
dim(mpg)## [1] 234 11summary(mpg)## manufacturer model displ year
## Length:234 Length:234 Min. :1.600 Min. :1999
## Class :character Class :character 1st Qu.:2.400 1st Qu.:1999
## Mode :character Mode :character Median :3.300 Median :2004
## Mean :3.472 Mean :2004
## 3rd Qu.:4.600 3rd Qu.:2008
## Max. :7.000 Max. :2008
## cyl trans drv cty
## Min. :4.000 Length:234 Length:234 Min. : 9.00
## 1st Qu.:4.000 Class :character Class :character 1st Qu.:14.00
## Median :6.000 Mode :character Mode :character Median :17.00
## Mean :5.889 Mean :16.86
## 3rd Qu.:8.000 3rd Qu.:19.00
## Max. :8.000 Max. :35.00
## hwy fl class
## Min. :12.00 Length:234 Length:234
## 1st Qu.:18.00 Class :character Class :character
## Median :24.00 Mode :character Mode :character
## Mean :23.44
## 3rd Qu.:27.00
## Max. :44.00- For Questions 2 to 8, we visit
flightsandweatherin thenycflights13dataset.
- How many observations and variables are there in the
flightsdataset? - Find out the meanings of
talinum,flight,carrier,dep_delayandarr_delayin theflightsdataset. - Find out the meanings of
visib,time_hour, andtempin theweatherdata set.
i.) There are 336776 observations in the flights
dataset.
dim(flights)## [1] 336776 19ii.) tailnum is the plane tail number,
flight is the flight number, carrier is the
two letter carrier abbreviation, dep_delay is the departure
delay, and arr_delay is the arrival delay.
iii.) visib is visibility in miles,
time_hour is date and hour of the recording, and
temp is temperature in F.
head(weather)## # A tibble: 6 × 15
## origin year month day hour temp dewp humid wind_dir wind_speed wind_gust
## <chr> <int> <int> <int> <int> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 EWR 2013 1 1 1 39.0 26.1 59.4 270 10.4 NA
## 2 EWR 2013 1 1 2 39.0 27.0 61.6 250 8.06 NA
## 3 EWR 2013 1 1 3 39.0 28.0 64.4 240 11.5 NA
## 4 EWR 2013 1 1 4 39.9 28.0 62.2 250 12.7 NA
## 5 EWR 2013 1 1 5 39.0 28.0 64.4 260 12.7 NA
## 6 EWR 2013 1 1 6 37.9 28.0 67.2 240 11.5 NA
## # … with 4 more variables: precip <dbl>, pressure <dbl>, visib <dbl>,
## # time_hour <dttm>- Extract the entries for Alaska Airlines from the
flightsdataset usingfilter()and%>%. Name the selected subset asalaska_flights. Furthermore, extract the weather data for EWR airport in January usingfilterand%>%, and name it asearly_january_weatherfromweather.
alaska_flights <- flights %>%
filter(carrier == "AS")
early_january_weather <- weather %>%
filter(origin == "EWR")- Create a scatterplot of
dep_delayandarr_delayin thealaska_flightsdata without using transparency. What do you find about the relationship between these two variable? What are some practical reasons for this relationship?
I notice that as departure delays increase, arrival delays increase as well. This is likely due to the fact that departure delays cause arrival delays, thus, when there are longer departure waits, there are longer arrival waits. I also noticed that most of the data points are concentrated around 0-50 minutes.
plot(alaska_flights$dep_delay, alaska_flights$arr_delay)
- Re-draw the plot in Part 4 with transparency set to be 0.2. Compare the two plot and explain your findings, if any.
Findings: after setting the transparency to 0.2, I can see that many of the data points are concentrated around 0-50 minutes
ggplot(alaska_flights, aes(dep_delay, arr_delay)) +
geom_point(alpha = 0.2)## Warning: Removed 5 rows containing missing values (geom_point).
- For the
early_january_weatherdata, create a linegraph withtime_houron thex-axis andtempon they-axis. What do you find from the plot?
I notice that temperature is the highest around July, which is accurate because that is when summer is. The lowest temperatures are around January, when winter is. The line graph allows me to see small oscillations in the data, which I assume represent the time of day. So at night it is cool and during the day it is warmer.
ggplot(early_january_weather, aes(time_hour, temp)) +
geom_line()
- Generate data from the following model. \[y_i = x_i\beta_1 + z_i\beta_2+x_i z_i \beta_3+
\epsilon_i, i = 1,\ldots, 100\] where \(\beta_1=1\), \(\beta_2=2\), \(\beta_3 =1\), and \(\epsilon_i \sim \mathcal{N}(0,1)\). For
\(x_i\) and \(z_i\), generate them as \[ x_i \sim\ {\rm uniform} \ (-2,2) \ {\rm
and}\ z_i =\tilde{\epsilon}_i,\] where \(\tilde{\epsilon}_i\sim
\mathcal{N}(0,0.5)\). Fit a linear regression with \(y\) as the outcome and \(x\) as the covariate (i.e., without \(z\)) using
lm(). Report the estimated coefficients.
The estimated coefficient for the intercept is 0.954 and the estimated coefficient for xi is 1.222.
set.seed(1)
b1 <- 1
b2 <- 2
b3 = 1
ei <- runif(100, min = 0, max = 1)
xi <- runif(100, min = -2, max = 2)
ei_tilda <- runif(100, min = 0, max= 0.5)
zi <- ei_tilda
# regression model
yi <- xi*b1 + zi*b2 + xi*zi*b3 + ei
# estimated coefficients
lm(yi ~ xi)##
## Call:
## lm(formula = yi ~ xi)
##
## Coefficients:
## (Intercept) xi
## 0.954 1.222- Wrap the code in Part 7 into a function, and run a simulation of 5000 instances to obtain the estimated coefficients. Draw a histogram of the estimated coefficients \(\hat{\beta}_1\)s of \(x\) from your simulation, add a vertical line (solid, black) to represent the simulation mean, and add another vertical line (dashed, red) to represent the true value (i.e., 1). Report your findings (hint: you may want to review unbiasedness).
FINISH
regression <- function() {
b1 <- 1
b2 <- 2
b3 = 1
ei <- runif(100, min = 0, max = 1)
xi <- runif(100, min = -2, max = 2)
ei_tilda <- runif(100, min = 0, max= 0.5)
zi <- ei_tilda
# regression model
yi <- xi*b1 + zi*b2 + xi*zi*b3 + ei
# estimated coefficients
lm(yi ~ xi)
}
coef_dist <- replicate(5000, summary(regression())$coefficients[2, 1])
hist(coef_dist)
abline(v = c(mean(coef_dist), 1), col = c('black', 'red'), lwd = 2, lty = "dashed")
- Report the simulation mean from Part 8. Then run another simulation under the same setting to verify that your simulation is reproducible. Specifically, the new simulation mean should be identical to the mean from Part 8.
The simulation mean from Part 8 was 1.25. When I reproduce the simulation, I get around the same mean at ~1.249.
mean(replicate(5000, summary(regression())$coefficients[2, 1]))## [1] 1.250421- Generate data from a similar model.
\[y_i = x_i\beta_1 + z_i\beta_2+\epsilon_i, i = 1,\ldots, 100\] where \(\beta_1=1\), \(\beta_2=2\), and \(\epsilon_i \sim \mathcal{N}(0,1)\). For \(x_i\) and \(z_i\), generate them as \[ x_i \sim\ {\rm uniform} \ (-2,2) \ {\rm and}\ z_i = \gamma_1 x_i+\gamma_0+\tilde{\epsilon}_i,\] where \(\gamma_1=1\), \(\gamma_0=1\)and \(\tilde{\epsilon}_i\sim \mathcal{N}(0,0.5)\). Suppose we fit a regression model with \(y\) as the outcome and \(x\) as the covariate (i.e., without \(z\)). Examine whether, in this model, the estimator for the coefficient of \(x\) is unbiased using simulation.
b1 <- 1
b2 <- 2
ei <- runif(100, 0, 1)
xi <- runif(100, -2, 2)
y1 <- 1
y0 <- 1
ei_tilda <- runif(100, 0, 0.5)
zi <- y1*xi+ y0 + ei_tilda
yi <- xi*b1 + zi*b2 + ei
lm(yi ~ xi)##
## Call:
## lm(formula = yi ~ xi)
##
## Coefficients:
## (Intercept) xi
## 3.069 2.976Acknowledgement
Failing to acknowledge any non-original efforts will be counted as plagiarism. This incidence will be reported to the Student Judicial Affairs.
If you use generative AI to solve any questions, please provide your instructions, conversations, and prompts here.
Session information
sessionInfo()## R version 4.2.1 (2022-06-23)
## Platform: x86_64-apple-darwin17.0 (64-bit)
## Running under: macOS Big Sur ... 10.16
##
## Matrix products: default
## BLAS: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRblas.0.dylib
## LAPACK: /Library/Frameworks/R.framework/Versions/4.2/Resources/lib/libRlapack.dylib
##
## locale:
## [1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
##
## attached base packages:
## [1] stats graphics grDevices utils datasets methods base
##
## other attached packages:
## [1] nycflights13_1.0.2 scales_1.2.1 forcats_0.5.2 stringr_1.4.1
## [5] dplyr_1.0.10 purrr_0.3.4 readr_2.1.2 tidyr_1.2.1
## [9] tibble_3.1.8 ggplot2_3.3.6 tidyverse_1.3.2
##
## loaded via a namespace (and not attached):
## [1] lubridate_1.8.0 assertthat_0.2.1 digest_0.6.33
## [4] utf8_1.2.2 R6_2.5.1 cellranger_1.1.0
## [7] backports_1.4.1 reprex_2.0.2 evaluate_0.23
## [10] httr_1.4.4 highr_0.9 pillar_1.8.1
## [13] rlang_1.1.2 googlesheets4_1.0.1 readxl_1.4.1
## [16] rstudioapi_0.14 jquerylib_0.1.4 rmarkdown_2.16
## [19] labeling_0.4.2 googledrive_2.0.0 munsell_0.5.0
## [22] broom_1.0.1 compiler_4.2.1 modelr_0.1.9
## [25] xfun_0.33 pkgconfig_2.0.3 htmltools_0.5.4
## [28] tidyselect_1.1.2 fansi_1.0.3 crayon_1.5.1
## [31] tzdb_0.3.0 dbplyr_2.2.1 withr_2.5.0
## [34] grid_4.2.1 jsonlite_1.8.7 gtable_0.3.1
## [37] lifecycle_1.0.4 DBI_1.1.3 magrittr_2.0.3
## [40] cli_3.6.1 stringi_1.7.8 cachem_1.0.6
## [43] farver_2.1.1 fs_1.5.2 xml2_1.3.3
## [46] bslib_0.4.0 ellipsis_0.3.2 generics_0.1.3
## [49] vctrs_0.4.1 tools_4.2.1 glue_1.6.2
## [52] hms_1.1.2 fastmap_1.1.0 yaml_2.3.5
## [55] colorspace_2.0-3 gargle_1.2.1 rvest_1.0.3
## [58] knitr_1.40 haven_2.5.1 sass_0.4.2