Part I: Defining & Plotting Datasets

Find pre-loaded datasets in R

data()

Cross Sectional (cars)

This dataset describes the speed at which cars were travelling and the distance it took them to come to a stop at said speed. The data was collected in the 1920s and used 50 separate observations.

speed = speed (mph)

dist = distance (ft)

col = “blue”)

?cars #find more info on dataset
data("cars") #Loads the dataset for analysis
cars #View dataset
##    speed dist
## 1      4    2
## 2      4   10
## 3      7    4
## 4      7   22
## 5      8   16
## 6      9   10
## 7     10   18
## 8     10   26
## 9     10   34
## 10    11   17
## 11    11   28
## 12    12   14
## 13    12   20
## 14    12   24
## 15    12   28
## 16    13   26
## 17    13   34
## 18    13   34
## 19    13   46
## 20    14   26
## 21    14   36
## 22    14   60
## 23    14   80
## 24    15   20
## 25    15   26
## 26    15   54
## 27    16   32
## 28    16   40
## 29    17   32
## 30    17   40
## 31    17   50
## 32    18   42
## 33    18   56
## 34    18   76
## 35    18   84
## 36    19   36
## 37    19   46
## 38    19   68
## 39    20   32
## 40    20   48
## 41    20   52
## 42    20   56
## 43    20   64
## 44    22   66
## 45    23   54
## 46    24   70
## 47    24   92
## 48    24   93
## 49    24  120
## 50    25   85
str(cars) #Summarize data used
## 'data.frame':    50 obs. of  2 variables:
##  $ speed: num  4 4 7 7 8 9 10 10 10 11 ...
##  $ dist : num  2 10 4 22 16 10 18 26 34 17 ...
plot(cars$speed, cars$dist,
     main = "Distance to Stop",
     ylab = "Distance (ft)",
     xlab = "Speed (mph)",
     pch = 19, #changes symbol used
     col = "blue")

As can be seen from the dataset and graph. The data collected contains 50 observations from around the same time period. Thus, it must be cross-sectional data.

Time Series (airmiles)

As per R’s provided description, the dataset describes “The revenue passenger miles flown by commercial airlines in the United States for each year from 1937 to 1960.”

?airmiles
airmiles
## Time Series:
## Start = 1937 
## End = 1960 
## Frequency = 1 
##  [1]   412   480   683  1052  1385  1418  1634  2178  3362  5948  6109  5981
## [13]  6753  8003 10566 12528 14760 16769 19819 22362 25340 25343 29269 30514

Since this is not in a data frame, we must convert it to one. The method for this is as follows:

data("airmiles")
df <- data.frame(Year = seq(1937, 1960), Airmiles = airmiles)
df
##    Year Airmiles
## 1  1937      412
## 2  1938      480
## 3  1939      683
## 4  1940     1052
## 5  1941     1385
## 6  1942     1418
## 7  1943     1634
## 8  1944     2178
## 9  1945     3362
## 10 1946     5948
## 11 1947     6109
## 12 1948     5981
## 13 1949     6753
## 14 1950     8003
## 15 1951    10566
## 16 1952    12528
## 17 1953    14760
## 18 1954    16769
## 19 1955    19819
## 20 1956    22362
## 21 1957    25340
## 22 1958    25343
## 23 1959    29269
## 24 1960    30514

Now we can use a graph and some other code to summarize the data

str(airmiles)
##  Time-Series [1:24] from 1937 to 1960: 412 480 683 1052 1385 ...
ts.plot(df$Airmiles,
        main = "Trend in Airmiles Over Time",
        ylab = "Airmiles",
        type = "b",
        col = "blue")

As we can see, the dataset is already described as a time series. Additionally, the dataset describes how one individual trended from 1937 to 1960

Part II: Additional Questions

Q1: What is covariance? What is variance?

Covariance:

Covariance is the statistical measure that quantifies how two random variables change together. It indicates whether there is a positive, negative, or no relationship between the variations in two sets of data points.

Variance:

Variance is the statistical measure that quantifies the degree of spread or dispersion in a set of data points. It represents how individual data points deviate from the mean of the data set.

Q2: Why does cov(y,x)/variance(x) give slope estimate?

As mentioned earlier, cov(y,x) measures how y and x change together and var(x) measures the spread of x. By dividing covariance over variance, we scale the changes in the two variables with a consistent spread of the values of the independent variable.

Q3: Show Q2 in R

We’ll demonstrate this using the cars dataset from earlier. If correct, Cov(speed,dist)/Var(speed) will equal\(\beta_1\) from the regression

#Find the covariance
?cov
covariance <- cov(cars$speed,cars$dist)
cat("Covariance:", covariance)
## Covariance: 109.9469
#Find the variance
variance <- var(cars$speed)
cat("Variance:", variance)
## Variance: 27.95918
#Find beta1
beta1 <-covariance/variance
cat("Slope Estimate:", beta1)
## Slope Estimate: 3.932409
#Run regression
model <- lm(dist ~ speed, data = cars)
model
## 
## Call:
## lm(formula = dist ~ speed, data = cars)
## 
## Coefficients:
## (Intercept)        speed  
##     -17.579        3.932

We can see that the slope estimate (beta1) equals the coefficient for speed from the regression. This shows that this proves it is the correct slope estimator in this instance.