Question

dependent variable is passengers
r - squared = 71.1% s = 104330
variable coefficient
constant 282584
year - 1990 59704.4
a) interpret the slope and intercept of the model.
b) what does the value of r² say about the model?
c) interpret sₑ in this context.
d) would you use this model to predict the numbers of passengers in 2010 (yearssince1990 = 20) explain.
e) there’s a point near the middle of this time span with a large negative residual. can you explain this outlier?
part ii
we updated the data through 2016 for this edition, and here is the scatterplot for data through 2016:
passengers
1,200,000
800,000
400,000
1990 1995 2000 2005 2010 2015
year
f) if you had used the model for the data through 2006 to predict the number of passengers in 2010, how well would it have worked? why?
g) explain how this shows the danger of extrapolating. what are possible reasons for the sudden decline in airline travel in about 2008?

1. tracking hurricanes 2015 in a previous chapter, we saw data

Explanation:

Step1: Interpret slope and intercept

The regression equation is $\text{Passengers}=282584 + 59704.4\times(\text{Year - 1990})$. The intercept 282584 represents the predicted number of passengers when Year - 1990 = 0 (i.e., in 1990). The slope 59704.4 means that for each additional year since 1990, the number of passengers is predicted to increase by 59704.4.

Step2: Interpret $R^{2}$

$R - squared=71.1\%$. This indicates that about 71.1% of the variation in the number of passengers can be explained by the linear relationship with the year since 1990. So, the model accounts for a substantial but not all of the variability in the data.

Step3: Interpret $s_{e}$

The value of $s = 104330$. This is the standard error of the estimate. It represents the standard deviation of the residuals. On average, the observed number of passengers deviates from the predicted number by about 104330.

Step4: Evaluate prediction for 2010

For 2010, YearsSince1990 = 20. $\text{Passengers}=282584+59704.4\times20=282584 + 1194088=1476672$. However, looking at the scatter - plot, there seems to be a change in the trend around 2008. So, it may not be reliable to use this model for prediction as the relationship may no longer be linear after 2008.

Step5: Explain outlier

A large negative residual means the observed number of passengers is much lower than the predicted number at that point. This could be due to unforeseen events such as a major economic recession, a significant airline - related accident, or a new travel - inhibiting regulation in that year.

Step6: Evaluate 2006 model for 2010 prediction

Using the model based on data through 2006 to predict for 2010 would likely not work well. The scatter - plot shows a change in trend around 2008. The model based on pre - 2008 data would not account for the decline in passengers starting around 2008.

Step7: Explain danger of extrapolation

Extrapolation is dangerous because the relationship between the year and the number of passengers may change over time. Possible reasons for the sudden decline in airline travel around 2008 could be the global financial crisis of 2008, which reduced consumers' disposable income for travel, or the emergence of new travel - limiting factors such as increased security measures that made air travel less convenient.

Answer:

a) Intercept: Predicted passengers in 1990 is 282584. Slope: Passengers increase by 59704.4 per year since 1990.
b) The model explains 71.1% of the variation in passengers.
c) On average, observed passengers deviate from predicted by 104330.
d) No, due to change in trend around 2008.
e) Could be due to economic, accident - related, or regulatory events.
f) Not well, as trend changed around 2008.
g) Relationship can change over time. Reasons for 2008 decline: financial crisis, increased security measures.