The table for part (a) shows distances between selected...

The table for part (a) shows distances between selected cities and the cost of a business class train ticket for travel between these cities. a. Calculate the correlation coefficient for the data shown in the table by using a computer or statistical calculator. $$ \begin{array}{|c|c|} \hline \text { Distance (in miles) } & \text { Cost (in \$) } \\ \hline 439 & 281 \\ \hline 102 & 152 \\ \hline 215 & 144 \\ \hline 310 & 293 \\ \hline 406 & 281 \\ \hline \end{array} $$ b. The table for part (b) shows the same information, except that the distance was converted to kilometers by multiplying the number of miles by $1.609$. What happens to the correlation when the numbers are multiplied by a constant? $$ \begin{array}{|c|c|} \hline \text { Distance (in kilometers) } & \text { Cost } \\ \hline 706 & 281 \\ \hline 164 & 152 \\ \hline 346 & 144 \\ \hline 499 & 293 \\ \hline 653 & 281 \\ \hline \end{array} $$ c. Suppose a surcharge is added to every train ticket to fund track maintenance. A fee of $$\$ 20$$ is added to each ticket, no matter how long the trip is. The following table shows the new data. What happens to the correlation coefficient when a constant is added to each number? $$ \begin{array}{|c|c|} \hline \text { Distance (in miles) } & \text { Cost (in \$) } \\ \hline 439 & 301 \\ \hline 102 & 172 \\ \hline 215 & 164 \\ \hline 310 & 313 \\ \hline 406 & 301 \\ \hline \end{array} $$

The table for part (a) shows distances between selected cities and the cost of a business class train ticket for travel between these cities.
a. Calculate the correlation coefficient for the data shown in the table by using a computer or statistical calculator.
$$
\begin{array}{|c|c|}
\hline \text { Distance (in miles) } & \text { Cost (in \$) } \\
\hline 439 & 281 \\
\hline 102 & 152 \\
\hline 215 & 144 \\
\hline 310 & 293 \\
\hline 406 & 281 \\
\hline
\end{array}
$$
b. The table for part (b) shows the same information, except that the distance was converted to kilometers by multiplying the number of miles by $1.609$. What happens to the correlation when the numbers are multiplied by a constant?
$$
\begin{array}{|c|c|}
\hline \text { Distance (in kilometers) } & \text { Cost } \\
\hline 706 & 281 \\
\hline 164 & 152 \\
\hline 346 & 144 \\
\hline 499 & 293 \\
\hline 653 & 281 \\
\hline
\end{array}
$$
c. Suppose a surcharge is added to every train ticket to fund track maintenance. A fee of $$\$ 20$$ is added to each ticket, no matter how long the trip is. The following table shows the new data. What happens to the correlation coefficient when a constant is added to each number?
$$
\begin{array}{|c|c|}
\hline \text { Distance (in miles) } & \text { Cost (in \$) } \\
\hline 439 & 301 \\
\hline 102 & 172 \\
\hline 215 & 164 \\
\hline 310 & 313 \\
\hline 406 & 301 \\
\hline
\end{array}
$$

Key Concepts

Correlation Coefficient

This is a statistical measure that quantifies the strength and direction of a linear relationship between two numerical variables. It is typically computed as the covariance of the two variables divided by the product of their standard deviations. The resulting value ranges from -1 to 1, where values close to 1 or -1 indicate strong linear relationships (positive or negative, respectively), and values near 0 indicate little or no linear relationship.

Multiplying by a Constant (Scaling Transformation)

Multiplying all values of a variable by a constant is known as a scaling transformation. When both variables are scaled, the correlation coefficient remains unchanged because the linear relationship's relative strength and direction are preserved. Although the actual data values change, the standardized measure of linear association is invariant under such scaling.

Adding a Constant (Translation Transformation)

Adding a constant to every value of a variable is known as a translation transformation. This adjustment shifts the data without affecting the spread or the relative distances between data points. Since the correlation coefficient is based on standardized deviations, adding a constant to one or both variables does not change the correlation value. This property underscores that correlation is a measure of the pattern of co-variation, independent of shifts in the location of the data.

Transcript

00:01 Okay, so here, the given data is for the distance between the cities and the cost of the train to get to travel between the cities.

00:09 So we have that our x here is going to be our distance in mile and the y is going to be our cost in dollars.

00:14 We want to find the correlation coefficient between the distance and the cost.

00:19 The correlation coefficient r is equal to the sum of each zx times zy divided by n minus 1, where zx is the z score for the x variable, and zy is the z score for the y.

00:30 Variable and n is the number of observed pairs in the given data so we find well for z x we have x minus x bar over the standard deviation so we define the mean for the x and y variable so that the mean for the x x bar is equal to the sum of all the x divided by n that's going to be equal to 1 ,472 divided by 5, giving x bar here equal to $294 .4 miles.

01:01 And then our mean for y, y bar, is equal to 1 ,151 divided by 5, which is going to be equal to $230 .20.

01:14 And then we can find a stamp with the deviation by taking each x minus x bar squared and each y minus y bar squared, summing all those together, and then we get the steady deviation for x is going to be equal to the square root of 76 ,229 .20 over 4.

01:36 Taking the square root gives us 138 .68 .68 miles, and the standard deviation for y is going to be equal to $75.

01:54 25 cents.

01:56 So then we get the correlation coefficient is going to be the sum of each xx times zy divided by n minus one, which is going to be equal to 3 .4386 divided by 5 minus 1.

02:14 So divided by 4 is going to be equal to 0 .86...

Please give Ace some feedback