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The distance (in kilometers) and price (in dollars) for one-way airline tickets from San Francisco to several cities are shown in the table. $$\begin{array}{|lcc|} \hline \text { Destination } & \text { Distance (km) } & \text { Price (\$) } \\\ \hline \text { Chicago } & 2960 & 229 \\ \hline \text { New York City } & 4139 & 299 \\ \hline \text { Seattle } & 1094 & 146 \\ \hline \text { Austin } & 2420 & 127 \\ \hline \text { Atlanta } & 3440 & 152 \\ \hline \end{array}$$ a. Find the correlation coefficient for this data using a computer or statistical calculator. Use distance as the \(x\) -variable and price as the \(y\) -variable. b. Recalculate the correlation coefficient for this data using price as the \(x\) -variable and distance as the \(y\) -variable. What effect does this have on the correlation coefficient? c. Suppose a $$\$ 50$$ security fee was added to the price of each ticket. What effect would this have on the correlation coefficient? d. Suppose the airline held an incredible sale, where travelers got a round- trip ticket for the price of a one-way ticket. This means that the distances would be doubled while the ticket price remained the same. What effect would this have on the correlation coefficient?

Step by step solution

01

Find the correlation coefficient using distance as the x-variable and price as the y-variable

To solve this, it is necessary to use the formula for the correlation coefficient, which for a set of data is given by the sum of xy divided by the square root of the sum of x squared, and the sum of y squared. However, due to the complexity and tedious nature of this task, a computer or statistical calculator is suggested. After inputting the data from the table, the correlation coefficient can be calculated.

02

Find the correlation coefficient using price as the x-variable and distance as the y-variable

Similar to the previous step, the correlation coefficient will again be calculated using a computer or a statistical calculator. Note that swapping the x and y variables does not affect the value of the correlation coefficient, it remains the same.

03

Find the correlation coefficient after adding a security fee

Adding a constant value to the y-variable (price) shifts all the points in the dataset upward. However, it does not change the spread of the data. This means that the correlation coefficient remains the same.

04

Find the correlation coefficient after doubling the distances

Doubling the x-variable (distance) stretches all the points of the dataset outwards, but does not impact the spread or the shape of the data. The correlation coefficient will therefore remain the same.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Statistical Calculator

Understanding and calculating the correlation coefficient, a measure of the strength and direction of the relationship between two variables, can be a complex task. To manage this, a statistical calculator comes into play. It's a tool designed to perform complex statistical operations, much like the one needed to find the correlation coefficient in our airline ticket price and distance example.

A statistical calculator simplifies the process by automating calculations. To determine the correlation coefficient using such a calculator, one would input the data points for distance and price. The machine then processes the values using the correlation formula, \( r = \frac{\sum (x-\bar{x})(y-\bar{y})}{\sqrt{\sum (x-\bar{x})^2 \sum (y-\bar{y})^2}} \) where \( \bar{x} \) and \( \bar{y} \) are the means of the x and y variables, respectively. This results in an efficient and error-free way to find the correlation coefficient compared to manual calculations.

Variables in Statistics

In statistics, variables are entities that we measure or observe. Two core types are relevant to correlation: dependent variables and independent variables. In our exercise, the distance could be considered an independent variable (\(x\)) as it does not depend on the ticket price. Conversely, the price could be viewed as the dependent variable (\(y\)), potentially influenced by the distance.

When calculating the correlation coefficient, changing the roles of the variables, as seen in step 2 of the solution, does not alter the correlation coefficient's value. This is because correlation is non-directional, reflecting the mutual relationship without implying causation. However, understanding the role of variables is crucial in other statistical analyses like regression, where the dependent and independent variable distinction is central to the interpretation of the results.

Effect of Data Transformation

Data transformation is a critical concept in statistics that refers to any change in the dataset which might occur during preprocessing. Common transformations include scaling, shifting (adding a constant), and more complex operations. In our ticket price exercise, two transformations were discussed:

• Adding a constant security fee to the ticket price
• Doubling the distance traveled, as in a round-trip scenario

When a constant is added to all values of one variable (as in the security fee case), it shifts the data points on the graph vertically but does not change their spacing. Similarly, multiplying a variable by a constant will stretch or shrink all values proportionally. Despite these changes, the overall relationship between the variables persists, and therefore, the correlation coefficient remains unaffected. It's key to recognize that while these transformations alter the dataset, they do not influence the strength or direction of the association measured by the correlation coefficient.