We are going to explore regression by look at the airmiles data set.

data(airmiles)
head(df)

  year miles
1 1937   412
2 1938   480
3 1939   683
4 1940  1052
5 1941  1385
6 1942  1418

`geom_smooth()` using formula = 'y ~ x'

Here is the code in R that shows how we built our linear trend model. We can see that geom_smooth adds a nice addition in a trend line as well to see where the airmiles data is going.

ggplot(df, aes(x=year,y=miles))+
  geom_point()+
  geom_smooth(method="lm",se=TRUE) +
  labs(
    title="Linear Trend with 95% Confidence Band",
    x="Year",
    y="Passenger miles (millions)"
  )

Modeling airline passenger miles as a linear function of time

\[ \text{Miles}_t = \beta_0 + \beta_1 \cdot \text{Year}_t + \varepsilon_t,\quad \varepsilon_t \sim N(0, \sigma^2) \]

• Miles is measured in Millions
• \(\beta_0\): intercept
• \(\beta_1\): average yearly change in passenger miles
• (_t: random error term

Testing if airline passenger miles have increased over time. We can see that our null hypothesis is that airline passanger miles have not increased over time.

\[ H_0:beta_1=0 \qquad \text{vs.}\qquad H_A:\beta_1>0 \]

The test statistic for the slope is:

\[ t=\frac{\hat{\beta}_1-0}{SE(\hat{\beta}_1)} \]

• A small p-value provides evidence of a positve time trend -This corresponds to the slope shown in the regression plot