Today….

In WF ED 540, you will learn to: import data; explore data through processes for transformation, visualization, and modeling; and communicate data that were refined and analyzed through processes of explanation and documentation. A demonstration is offered of almost all of the ways you will learn to process data in WF ED 540.

Nature of Demonstration Offered

In this demonstration I use mtcars, a dataset available through the “datasets” package that accompanies the base R installation.

  • First, one way to refine this dataset is demonstrated (There are many ways, as with all of the processes I demonstrate with mtcars).

  • Then, some variables in mtcars are transformed in various ways using functions available in the R package called dplyr.

  • Next, vizualizations of some variables in mtcars are shown using functions available in the R package ggplot2.

  • Last, several simple statistical hypotheses are modeled with some variables in the mtcars dataset.

Explanation and documentation? The example of these processes in the data sciences is provided in this file, WFED540Overview_23Aug2065.Rmd, which is an RMarkdown file. RMarkdown is an authoring language that enables easy creation of dynamic documents, presentations, and reports from R. It combines the core syntax of markdown (an easy-to-write plain text format commonly applied in data science) with embedded “chunks” of R code that are run so their output can be included in a final document. You receive instruction in WF ED 540 in the creation of RMarkdown files and for displaying their output as HTML files.

You, too, will be able to run this RMarkdown file, WFED540Overview_27Aug2015.Rmd, so that you can examine the scope of what we will learn and practice throughout WF ED 540. You receive in a Piazza note a URL with a link for you to download WFED540Overview_27Aug2015.Rmd as a .zip file. Unzip this file. Double-click on the unzipped file to open it in RStudio. Press the “Knit HTML” button. “Kniiting” this file produced a full formatted web page for you to examine.

Note: This is a Tour, Not Instruction

This report is not meant to instruct you in the use of R or RStudio. It merely is an introductory tour of almost all of the R and RStudio capabilities that you wil develop in WF ED 540.

Refining the dataset mtcars

Structure of mtcars

To demonstrate processes in R, I examine mtcars, a dataset with small number of observations and variables. The data in mtcars was extracted from the 1974 Motor Trend US magazine, and comprises fuel consumption and 10 aspects of automobile design and performance for 32 automobiles (1973–74 models). Said in another way, the data set, mtcars, has 32 observations (cars) with measures for 11 variables (aspects of design and performance).

During Class Meeting 1 you were provided with a printed copy of a description and listing of the 32-observation-by-11-variable mtcars dataset to which you can refer throughout this demonstration.

Loading mtcars in RStudio

By entering the name of the dataset in the R console quadrant or in the source quadrant will print the dataset in the console:

mtcars
##                      mpg cyl  disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4           21.0   6 160.0 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag       21.0   6 160.0 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710          22.8   4 108.0  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive      21.4   6 258.0 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout   18.7   8 360.0 175 3.15 3.440 17.02  0  0    3    2
## Valiant             18.1   6 225.0 105 2.76 3.460 20.22  1  0    3    1
## Duster 360          14.3   8 360.0 245 3.21 3.570 15.84  0  0    3    4
## Merc 240D           24.4   4 146.7  62 3.69 3.190 20.00  1  0    4    2
## Merc 230            22.8   4 140.8  95 3.92 3.150 22.90  1  0    4    2
## Merc 280            19.2   6 167.6 123 3.92 3.440 18.30  1  0    4    4
## Merc 280C           17.8   6 167.6 123 3.92 3.440 18.90  1  0    4    4
## Merc 450SE          16.4   8 275.8 180 3.07 4.070 17.40  0  0    3    3
## Merc 450SL          17.3   8 275.8 180 3.07 3.730 17.60  0  0    3    3
## Merc 450SLC         15.2   8 275.8 180 3.07 3.780 18.00  0  0    3    3
## Cadillac Fleetwood  10.4   8 472.0 205 2.93 5.250 17.98  0  0    3    4
## Lincoln Continental 10.4   8 460.0 215 3.00 5.424 17.82  0  0    3    4
## Chrysler Imperial   14.7   8 440.0 230 3.23 5.345 17.42  0  0    3    4
## Fiat 128            32.4   4  78.7  66 4.08 2.200 19.47  1  1    4    1
## Honda Civic         30.4   4  75.7  52 4.93 1.615 18.52  1  1    4    2
## Toyota Corolla      33.9   4  71.1  65 4.22 1.835 19.90  1  1    4    1
## Toyota Corona       21.5   4 120.1  97 3.70 2.465 20.01  1  0    3    1
## Dodge Challenger    15.5   8 318.0 150 2.76 3.520 16.87  0  0    3    2
## AMC Javelin         15.2   8 304.0 150 3.15 3.435 17.30  0  0    3    2
## Camaro Z28          13.3   8 350.0 245 3.73 3.840 15.41  0  0    3    4
## Pontiac Firebird    19.2   8 400.0 175 3.08 3.845 17.05  0  0    3    2
## Fiat X1-9           27.3   4  79.0  66 4.08 1.935 18.90  1  1    4    1
## Porsche 914-2       26.0   4 120.3  91 4.43 2.140 16.70  0  1    5    2
## Lotus Europa        30.4   4  95.1 113 3.77 1.513 16.90  1  1    5    2
## Ford Pantera L      15.8   8 351.0 264 4.22 3.170 14.50  0  1    5    4
## Ferrari Dino        19.7   6 145.0 175 3.62 2.770 15.50  0  1    5    6
## Maserati Bora       15.0   8 301.0 335 3.54 3.570 14.60  0  1    5    8
## Volvo 142E          21.4   4 121.0 109 4.11 2.780 18.60  1  1    4    2

Printing mtcars to the RStudio console

Printing a dataset like mtcars is suitable for viewing in the console. It contains few observations and variables. However, what about datasets that contain hundreds … thousands … hundreds of thousands of observations?

Another approach is to print in the console the first six lines of the dataset using the R function, head.

head(mtcars)
##                    mpg cyl disp  hp drat    wt  qsec vs am gear carb
## Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
## Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
## Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
## Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
## Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
## Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

Or, to print the last six lines.

tail(mtcars)
##                 mpg cyl  disp  hp drat    wt qsec vs am gear carb
## Porsche 914-2  26.0   4 120.3  91 4.43 2.140 16.7  0  1    5    2
## Lotus Europa   30.4   4  95.1 113 3.77 1.513 16.9  1  1    5    2
## Ford Pantera L 15.8   8 351.0 264 4.22 3.170 14.5  0  1    5    4
## Ferrari Dino   19.7   6 145.0 175 3.62 2.770 15.5  0  1    5    6
## Maserati Bora  15.0   8 301.0 335 3.54 3.570 14.6  0  1    5    8
## Volvo 142E     21.4   4 121.0 109 4.11 2.780 18.6  1  1    4    2

But, these printing approaches are not entirely satisfactory when there are many variables in a dataset. Many datasets you will examine in WF ED 540 contain thousands of variables.

Transforming mtcars using dplyr

Loading dplyr

We can load an R “package” (a special purpose library of code in R that accomplishes a bundle of computing tasks) called dplyr to provide some order to the mtcars data. We create a more orderly table of the mtcars dataset by executing what is called a tibble function, tbl_df, to transform mtcars into a new table of data, mtcars_df. (tibble….tbl_df….get it?)

library(dplyr)
## Warning: package 'dplyr' was built under R version 3.2.5
## 
## Attaching package: 'dplyr'
## The following objects are masked from 'package:stats':
## 
##     filter, lag
## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union
mtcars_df <- tbl_df(mtcars)

Exploring mtcars with dplyr

Now, a print of mtcars_df is more orderly set of columns (variables) and only the first 10 rows (observations…or cars):

mtcars_df
## # A tibble: 32 x 11
##      mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear  carb
## *  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1   21.0     6 160.0   110  3.90 2.620 16.46     0     1     4     4
## 2   21.0     6 160.0   110  3.90 2.875 17.02     0     1     4     4
## 3   22.8     4 108.0    93  3.85 2.320 18.61     1     1     4     1
## 4   21.4     6 258.0   110  3.08 3.215 19.44     1     0     3     1
## 5   18.7     8 360.0   175  3.15 3.440 17.02     0     0     3     2
## 6   18.1     6 225.0   105  2.76 3.460 20.22     1     0     3     1
## 7   14.3     8 360.0   245  3.21 3.570 15.84     0     0     3     4
## 8   24.4     4 146.7    62  3.69 3.190 20.00     1     0     4     2
## 9   22.8     4 140.8    95  3.92 3.150 22.90     1     0     4     2
## 10  19.2     6 167.6   123  3.92 3.440 18.30     1     0     4     4
## # ... with 22 more rows

Now that mtcars has been transformed, the result of the transformation, mtcars_df, is described in R as a dataframe rather than a dataset.

Also, note that the size of the table of data also is printed (32 rows by 11 columns….or, in our case, 32 cars by 11 aspects of performance).

We can use the dplyr function, glimpse, to view variable names, the type of variables (in this case “dbl” meaning numeric data), and a sample of some of the data points in mtcars_df:

glimpse(mtcars_df)
## Observations: 32
## Variables: 11
## $ mpg  <dbl> 21.0, 21.0, 22.8, 21.4, 18.7, 18.1, 14.3, 24.4, 22.8, 19....
## $ cyl  <dbl> 6, 6, 4, 6, 8, 6, 8, 4, 4, 6, 6, 8, 8, 8, 8, 8, 8, 4, 4, ...
## $ disp <dbl> 160.0, 160.0, 108.0, 258.0, 360.0, 225.0, 360.0, 146.7, 1...
## $ hp   <dbl> 110, 110, 93, 110, 175, 105, 245, 62, 95, 123, 123, 180, ...
## $ drat <dbl> 3.90, 3.90, 3.85, 3.08, 3.15, 2.76, 3.21, 3.69, 3.92, 3.9...
## $ wt   <dbl> 2.620, 2.875, 2.320, 3.215, 3.440, 3.460, 3.570, 3.190, 3...
## $ qsec <dbl> 16.46, 17.02, 18.61, 19.44, 17.02, 20.22, 15.84, 20.00, 2...
## $ vs   <dbl> 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, ...
## $ am   <dbl> 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, ...
## $ gear <dbl> 4, 4, 4, 3, 3, 3, 3, 4, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, ...
## $ carb <dbl> 4, 4, 1, 1, 2, 1, 4, 2, 2, 4, 4, 3, 3, 3, 4, 4, 4, 1, 2, ...

The dataset, mtcars, and the dataframe created from it, mtcars_df, do not contain any missing data. It is quite common in practical research settings to have missing data for some cases (in mtcars_df, cars) within variables (in mtcars_df, aspects of performance). In WF ED 540 you will learn how to handle missing data in data modeling.

Summarising mtcars_df variables using dplyr functions

Rarely do researchers analyze the raw data that they collect. The R package, dplyr, makes reshaping data for analysis a relatively easy task.

It is common to filter a subset of rows in a data table according to criteria (dplyr’s filter function). Or, perhaps to select only some variables (dplyr’s select function).

Often, researchers wish to create new variables from existing variables (dplyr’s mutate function). And, statistical summaries of variables are desired (dplyr’s summarise function; note the spelling!).

The important and efficient use of dplyr is to use a small set of verbs to reshape data. Among the dplyr verbs you will learn to use to reshape data tables are:

  • filter

  • select

  • mutate

  • summarise

Provided are some examples of use of these verbs with the dataset, mtcars_df. Compare the transformations made with the raw data shown in the information sheet about mtcars that you received at the beginning of the first class meeting.

Filter mtcars

Here is a filter of only the cars that have a straight engine:

filter(mtcars_df, vs == 1)
## # A tibble: 14 x 11
##      mpg   cyl  disp    hp  drat    wt  qsec    vs    am  gear  carb
##    <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1   22.8     4 108.0    93  3.85 2.320 18.61     1     1     4     1
## 2   21.4     6 258.0   110  3.08 3.215 19.44     1     0     3     1
## 3   18.1     6 225.0   105  2.76 3.460 20.22     1     0     3     1
## 4   24.4     4 146.7    62  3.69 3.190 20.00     1     0     4     2
## 5   22.8     4 140.8    95  3.92 3.150 22.90     1     0     4     2
## 6   19.2     6 167.6   123  3.92 3.440 18.30     1     0     4     4
## 7   17.8     6 167.6   123  3.92 3.440 18.90     1     0     4     4
## 8   32.4     4  78.7    66  4.08 2.200 19.47     1     1     4     1
## 9   30.4     4  75.7    52  4.93 1.615 18.52     1     1     4     2
## 10  33.9     4  71.1    65  4.22 1.835 19.90     1     1     4     1
## 11  21.5     4 120.1    97  3.70 2.465 20.01     1     0     3     1
## 12  27.3     4  79.0    66  4.08 1.935 18.90     1     1     4     1
## 13  30.4     4  95.1   113  3.77 1.513 16.90     1     1     5     2
## 14  21.4     4 121.0   109  4.11 2.780 18.60     1     1     4     2

Notice that 14 of the 32 cars meet this filtering criterion (see “Source: local data frame [14 x 11]”).

The filter verb has two parameters: the dataset to be filtered (mtcars_df); the filtering criterion (vs == 1, where “==” means “the same as”).

Select mtcars variables

Suppose I wish to select only mtcars_df variables that contain measures of weight (in pounds/1,000) and number of forward gears:

select(mtcars_df, wt, gear)
## # A tibble: 32 x 2
##       wt  gear
## *  <dbl> <dbl>
## 1  2.620     4
## 2  2.875     4
## 3  2.320     4
## 4  3.215     3
## 5  3.440     3
## 6  3.460     3
## 7  3.570     3
## 8  3.190     4
## 9  3.150     4
## 10 3.440     4
## # ... with 22 more rows

Notice that the selection retains all 32 cars, but only two variables (see “Source: local data frame [32 x 2]”)

Mutate mtcars

Now, suppose I wish to mutate mtcars_df by calculating a new variable from disp, the engine displacment in cubic inches: disp_l, the displacement in liters:

displace <- mutate(mtcars, displ_l = disp / 61.0237)
select (displace, disp, displ_l)
##     disp  displ_l
## 1  160.0 2.621932
## 2  160.0 2.621932
## 3  108.0 1.769804
## 4  258.0 4.227866
## 5  360.0 5.899347
## 6  225.0 3.687092
## 7  360.0 5.899347
## 8  146.7 2.403984
## 9  140.8 2.307300
## 10 167.6 2.746474
## 11 167.6 2.746474
## 12 275.8 4.519556
## 13 275.8 4.519556
## 14 275.8 4.519556
## 15 472.0 7.734700
## 16 460.0 7.538055
## 17 440.0 7.210313
## 18  78.7 1.289663
## 19  75.7 1.240502
## 20  71.1 1.165121
## 21 120.1 1.968088
## 22 318.0 5.211090
## 23 304.0 4.981671
## 24 350.0 5.735477
## 25 400.0 6.554830
## 26  79.0 1.294579
## 27 120.3 1.971365
## 28  95.1 1.558411
## 29 351.0 5.751864
## 30 145.0 2.376126
## 31 301.0 4.932510
## 32 121.0 1.982836

I created a new variable, displ_l, by converting cubic inches to liters (there are 61.0237 cubic inches in a liter). I created a new data table, displace, from which I, then, selected only disp and displ_l, the displacement in cubic inches and liters, respectively.

Summarize mtcars

Last, let’s calculate some summary statistics for a few variables in the new data I created, displace. Calculated are the means for disp and displ_l:

summarise(displace, count=n(), mean_cu_in = mean(disp), mean_ltr=mean(displ_l))
##   count mean_cu_in mean_ltr
## 1    32   230.7219 3.780857

I added the function, count, to return the number of cars in mtcars_df. The means are the sum of all disp or displ_l divided by the number of cars.

Visualizing mtcars_df variables using ggplot2

As indicated in a white paper by Fusion Charts,

We visualize information to meet a very basic need – to tell a story. It’s one of the most primitive forms of communication known to man, having its origins in cave drawings dated as early as 30,000 B.C., even before written communication, which emerged in 3,000 B.C. Vision is the single most important faculty we use to communicate information.

A R package, ggplot2, provides a grammar for commands that create visualizations of data.

The concept behind ggplot2 divides plot into three different fundamental parts: Plot = data + Aesthetics + Geometry. The principal components of every plot can be defined as follow:

Simple bivariate plot

I wish to plot the relationship between the weight of a car and its mileage. Make sure you have ggplot2 installed. We load the ggplot2 library. Then, I use the ggplot2 code to creates a scatterplot of car miles per gallon by car weight.

In this R code, a plot is created using a dataframe (mtcars_df) + aesthetics (the x and y axes) + a geometry (geom_points, which plots points):

# Make sure you have installed the ggplot2 package.
library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.2.4
  ggplot(mtcars_df, aes(x=wt, y=mpg)) + 
   geom_point()

Plotted on the graph are the weight (wt) and lines per gallon (mpg) for every car. The joint occurence of wt and mpg is what is called a Cartesian coordinate and is show by a point for each car.

Change size of points on plot

But, hmmm, the points are small, aren’t they? So, let’s change the size of the points by modifying geom_point() to geom_point(size=5).

  ggplot(mtcars_df, aes(x=wt, y=mpg)) + 
   geom_point(size=5)

Change the shape of points on plot

Let’s change the plot so that the shape of points is now a diamond (shape=23), the size is larger (size=8_), the color is red (color=“red”), and the diamonds are not filled (fill=NA:

   ggplot(mtcars_df, aes(x=wt, y=mpg)) + 
   geom_point(size=8, shape=23, color="red", fill=NA)

Draw a line through points on plot

How about adding a line though black, round points (minus some of the other previously used plot charactertistics) that minimizes the distance between the line and the swarm of points?

  ggplot(mtcars_df, aes(x=wt, y=mpg)) +
  geom_point(size=4) +
  geom_smooth(method=lm, se=FALSE)

Now, change the line so that it follows the contour of the relationship between wt and mpg:

  ggplot(mtcars_df, aes(x=wt, y=mpg)) +
  geom_point(size=4) +
  geom_smooth(se=FALSE)

Change the color of the smooth line? Add color=“red” to the geom_smooth characteristic.

  ggplot(mtcars_df, aes(x=wt, y=mpg)) +
  geom_point(size=4) +
  geom_smooth(se=FALSE, color="red")

Label axes and add a title

Pretty. But, what about a title and labels for axes of the graph?

  ggplot(mtcars_df, aes(x=wt, y=mpg)) +
  geom_point(size=4) +
  geom_smooth(se=FALSE, color="red") +
  labs(title="Relationship Between Miles Per Gallon and Weight",
       x="Weight", y="Miles Per Gallon")

Add a third variable to the plot

Now, let’s plot wt and mpg by the number of cylinders (cyl), make points based on different numbers of cylinders different shapes and colors, and add a legend to the plot?

mtcars_df$cyl <- as.factor(mtcars_df$cyl) 
# Previous command changes the cyl variable from numeric 
# to a factor (a categorical variable).
 ggplot(mtcars_df, aes(x=wt, y=mpg, shape=cyl, color=cyl)) +
   geom_point(size=4) +
  labs(title="Relationship Between Miles Per Gallon and Weight",
       x="Weight", y="Miles Per Gallon")

Move the legend to the top of the plot

 ggplot(mtcars_df, aes(x=wt, y=mpg, shape=cyl, color=cyl)) +
   geom_point(size=4) +
  labs(title="Relationship Between Miles Per Gallon and Weight",
       x="Weight", y="Miles Per Gallon") +
   theme(legend.position="top")

Change the look and feel of the plot using themes

Change the “look and feel” of the plot I added theme_economist, a based on plot design for The Economist magazine, after installing and loading the package, “ggthemes.”

# Make sure you have installed the ggthemes package.
library(ggthemes)
## Warning: package 'ggthemes' was built under R version 3.2.5
mtcars_df$cyl <- as.factor(mtcars_df$cyl) 
 ggplot(mtcars_df, aes(x=wt, y=mpg, shape=cyl, color=cyl)) +
   geom_point(size=4) +
  labs(title="Relationship Between Miles Per Gallon and Weight",
       x="Weight", y="Miles Per Gallon") +
   theme(legend.position="top") + 
   theme_economist()

Some academic journals do not accept color graphics. This plot uses a grey scale color palette entirely:

 ggplot(mtcars_df, aes(x=wt, y=mpg, shape=cyl, color=cyl)) +
   geom_point(size=4) +
  labs(title="Relationship Between Miles Per Gallon and Weight",
       x="Weight", y="Miles Per Gallon") +
   theme(legend.position="top",
         axis.title=element_text(size=17),
         axis.text=element_text(size=13),
         plot.title=element_text(size=19)) + 
   scale_color_grey()

ggpplot2 actually is simple

I am able to change the graph just by changing simple ggplot2 properties. In WF ED 540, you will learn and practice making visualizations such as scatterplots, bar graphs, line graphs, histograms, regrssion lines, box plots, and other plot types.

Once you get the hang of it, plotting becomes simple because there is a simple grammar for ggplot2 that specifies the dataset to be plotted, the coordinate system applied to the plot, and the style of points plotted.

Statistical Modeling with Some Variables in mtcars_df

Simple t-test

Suppose I am interested in testing the null hypothesis that fuel efficiency, miles/gallon, is equal for cars with automatic and manual transmissions. There are observable differences in our 32-car sample. But, mtcars is just a sample from a population of cars. Can I infer that mileage is the same or different in the population of cars based on this sample of data.

To test this hypothesis, I structure a t-test of the differences between mean mileage by transmission type:

mpg.at <- mtcars_df[mtcars$am == 0,]$mpg
mpg.mt <- mtcars_df[mtcars$am == 1,]$mpg
t.test(mpg.at, mpg.mt) 
## 
##  Welch Two Sample t-test
## 
## data:  mpg.at and mpg.mt
## t = -3.7671, df = 18.332, p-value = 0.001374
## alternative hypothesis: true difference in means is not equal to 0
## 95 percent confidence interval:
##  -11.280194  -3.209684
## sample estimates:
## mean of x mean of y 
##  17.14737  24.39231

The mean mileage for cars with manual transmissions is approximatly 24, while for automatics is approximatly 17 – a 7 miles/gallon difference in the sample. According to my analysis, I am 95% confident that the actual mileage difference between automatic and manual transmissions is between 3 and 11 miles per gallon.

If I set the probability of Type 1 error (error made rejecting the null hypothesis when it should not be rejected) to 0.05, I can reject the null hypothesis of “no difference” because the probability is 0.0013 of observing a t-value of 3.76, with 18 degrees of freedom, if the null hypothesis is true.

Hey, now, that’s a mouthful. But, you will not only learn to talk that gibberish in WF ED 540, but you also will learn and practice to understand it!

Now, this whole thing in a graphic:

boxplot(mpg~am, data=mtcars_df, main ='Fig. 1. Fuel Efficiency',
        ylab='Miles per gallon',names=c("Automatic","Manual"),notch=FALSE, col=(c("gold","skyblue")))

Simple Regression

OK, now, I test this hypothesis in another way – through linear regression.

Linear regression fits a line a “best fit” through a plot of points formed by the intersection of coordinates formed by values of two variables. The equation of a regression line is y = a + bx, where y is a depndent variable (mpg in our case), x is an independent variable (type of transmission), a is an intercpt term, and b is the slope of the “best fit” rgression line.

So, in R:

fit_1var <- lm(mtcars$mpg ~ mtcars$am)
summary(fit_1var)$coefficients
##              Estimate Std. Error   t value     Pr(>|t|)
## (Intercept) 17.147368   1.124603 15.247492 1.133983e-15
## mtcars$am    7.244939   1.764422  4.106127 2.850207e-04
par(mfrow = c(1,2))
plot(mtcars$am, mtcars$mpg, main = 'Simple Linear Regression: MPG vs Transmission Type')
abline(fit_1var)

The slope of the regression line is appoximately 7, which is change in mpg for a change of one unit in am (the difference between automatic transmissions, coded “0”, and manual transmissions, coded “1”). This is the same result as for the t-test.

In WF ED 540, you learn and practice the fundamentals of statistical null hypothesis testing and estimation using t-tests, chi-square tests of independence, and simple linear regression.

Explanation and Documentation

This report is an example of explanation of findings of data analysis. Moreover, this report also is an example of documentation through use of RMarkdown.

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This document is hosted at http://rpubs.com/davidpassmore/105923.