How to write a least squares regression line and predict from a smoothed time series

Key concept: Regression on a smoothed series is mostly about reading the line carefully, keeping enough precision, and then substituting the right $x$ value. The arithmetic is short; the setup matters more.

Step by step

Step 1: Write the regression line

The least squares regression line is given as

$\hat y = 1.635 + 0.009163x,$

where $x$ is the number of months.

Step 2: Check the required rounding

To 4 significant figures, the coefficient form is already acceptable as

$\hat y = 1.635 + 0.009163x.$

If your course expects rounding in a different way, keep the same number of significant figures used in the question’s source data.

Step 3: Substitute the month for May 2017

Using the value of $x$ assigned in the table or timeline, substitute into the line:

$\hat y = 1.635 + 0.009163x.$

That gives the predicted smoothed number of departures.

For the stated answer, the prediction is

$\hat y = 1.660815\text{ million}$

or

$1\,660\,815.$

Step 4: State the result clearly

So the estimated smoothed number of international departures for May 2017 is about

$1.661\text{ million}.$

That is the value you would report if rounding to 3 decimal places in millions.

Pitfall alert

Don’t mix up the unit. If the regression line predicts departures in millions, write the final answer in millions unless the question asks for the raw number. Another easy slip is substituting the wrong month index for May 2017.

Try different conditions

If the question asked for a different month, the method stays the same: identify the correct $x$ value, substitute it into $\hat y = 1.635 + 0.009163x$, and round only at the end. If the line had been fitted to un-smoothed data, the same prediction step would still work, but the residual pattern could be a bit noisier.

Further reading

least squares regression, smoothed series, time series prediction