The following data represents nonstop flight times​ (in minutes) and​ one-way ticket prices​ (in dollars) for flying from City A to various cities on certain airlines. Use the given information to answer parts​ (a) through​ (e).
City
Time​ (minutes), t
Price​ (dollars),
P
City B
110
140
City C
260
199
City D
81
108
City G
126
130
City H
86
87
City K
169
175
City L
226
191
City M
155
147
City N
265
196
City P
64
93

The scatterplot above shows the relationship between flight times (in minutes) and ticket prices (in dollars) for the flights from City A to various cities. Each point represents one city, with the flight time on the x-axis and the price on the y-axis.

### (c) Does the scatterplot show a positive or negative linear​ trend?
The scatterplot appears to show a **positive linear trend**. As the flight time increases, the price tends to increase as well. Although there is some variability, the general trend indicates that longer flight times correspond to higher prices.

### (d) Use the​ least-squares method to determine the regression​ equation.
The regression equation can be written as:
\[
P = b_0 + b_1 t
\]
where $P$ is the predicted price, $t$ is the flight time, $b_0$ is the intercept, and $b_1$ is the slope of the line.

I'll calculate the regression line using the least-squares method.The regression equation based on the least-squares method is:

\[
P = 62.55 + 0.545 \cdot t
\]

Where:
- $62.55$ is the intercept, representing the predicted price when the flight time is zero (which is theoretical and doesn't occur in reality).
- $0.545$ is the slope, indicating that for each additional minute of flight time, the ticket price increases by approximately $0.55.

### (e) Predict the price for a flight time of 180 minutes.
Using the regression equation $P = 62.55 + 0.545 \cdot t$, we can substitute $t = 180$ to predict the price for a 180-minute flight.

Let me calculate that for you.The predicted price for a flight time of 180 minutes is approximately **$160.66**.

Would you like any further details or clarification on these results?

Here are five related questions to consider:
1. How would the scatterplot look if only shorter flights (under 150 minutes) were considered?
2. What impact does outlier data have on the regression line?
3. Could the relationship between time and price be better modeled by a non-linear trend?
4. How reliable is the prediction for a flight with a time far outside the observed range?
5. How can you calculate the strength of the correlation between time and price?

**Tip**: When interpreting a regression model, always check for outliers or influential points that could distort the trend.

The following data represents nonstop flight times​ (in minutes) and​ one-way ticket prices​ (in dollars) for flying from City A to various cities on certain airlines. Use the given information to answer parts​ (a) through​ (e).

Explore the linear regression analysis between nonstop flight times (in minutes) and one-way ticket prices (in dollars) for various cities. This problem uses statistical methods such as the least-squares method to derive a regression equation and predict flight prices based on travel time. Suitable for students in Grades 10-12.

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