The distance (in kilometers) and price (in dollars) for...

The distance (in kilometers) and price (in dollars) for one-way airline tickets from San Francisco to several cities are shown in the table. $$ \begin{array}{|lcc|} \hline \text { Destination } & \text { Distance }(\mathbf{k m}) & \text { Price (\$) } \\ \hline \text { Chicago } & 2960 & 229 \\ \hline \text { New York City } & 4139 & 299 \\ \hline \text { Seattle } & 1094 & 146 \\ \hline \text { Austin } & 2420 & 127 \\ \hline \text { Atlanta } & 3440 & 152 \\ \hline \end{array} $$ 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?

The distance (in kilometers) and price (in dollars) for one-way airline tickets from San Francisco to several cities are shown in the table.
$$
\begin{array}{|lcc|}
\hline \text { Destination } & \text { Distance }(\mathbf{k m}) & \text { Price (\$) } \\
\hline \text { Chicago } & 2960 & 229 \\
\hline \text { New York City } & 4139 & 299 \\
\hline \text { Seattle } & 1094 & 146 \\
\hline \text { Austin } & 2420 & 127 \\
\hline \text { Atlanta } & 3440 & 152 \\
\hline
\end{array}
$$
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?

Key Concepts

Invariance under Scaling

The correlation coefficient remains unchanged under positive scaling (multiplying by a constant) of one or both variables. Scaling the data by a constant factor (stretching or compressing the data) does not alter the linear relationship between the variables as long as the transformation is linear, meaning the correlation coefficient is unaffected by such multiplicative changes.

Invariance under Translation

A key property of the correlation coefficient is that it is invariant under translations; this means that adding a constant to every value of one of the variables (a vertical or horizontal shift) does not affect the correlation. The underlying relationships in the variances and covariances remain unchanged after such additive adjustments.

Correlation Coefficient

The correlation coefficient is a statistical measure that quantifies the strength and direction of a linear relationship between two quantitative variables. It takes values between -1 and 1, where values close to 1 or -1 indicate a strong linear relationship while values near 0 indicate weak or no linear relationship.

Symmetry of Correlation

The correlation coefficient is symmetric, meaning it is the same regardless of which variable is considered the independent variable (x) and which is considered the dependent variable (y). This property implies that swapping the roles of the variables does not change the value of the correlation coefficient.

Transcript

0:00 Hello.

00:01 So here we are trying to calculate the correlation coefficient between the distance and price.

00:06 So the correlation coefficient, r, is going to be calculated as the sum of xx times zy divided by n minus 1.

00:15 Now z x is going to be the z score for the x variable, and zy is going to be the z score for the y variable.

00:22 So the value of z x is calculated as x minus x bar divided by the static deviation.

00:28 So we have to first find the mean of the x and y's.

00:33 So we get that x bar.

00:36 The mean is going to be equal to the sum of all the x is divided by n.

00:41 That is going to be equal to 14 ,053 divided by five.

00:51 So x bar here is equal to 2 ,810 .6.

00:55 And then we get that y bar is the sum of the ys divided by n.

01:00 That's going to be 953 divided by 5, which is going to be equal to 190 .6.

01:09 And then we can take each of the y minus y bar squared and take the sum to find the standard deviation.

01:19 So the standard deviation we know for x is going to be equal to, well, the sum of each x minus x bar squared divided by n minus 1 and then taking the square root.

01:33 So that's going to be equal to 1149 .17.

01:41 And then likewise, the standard deviation of y is equal to the sum of each y minus y bar squared by n minus 1, taking the square root, giving us 72 .02.

01:53 So then to find the correlation coefficient r is going to be equal to the sum of xx times z, which is going to give us 2 .7409.

02:09 And then we divide that by n minus 1.

02:11 We divide that by 4, giving us the correlation coefficient is 0 .69.

02:18 So therefore, the correlation coefficient between the distance and the price is going to be 0 .69.

02:25 Then in b, we want to calculate the correlation coefficient using the price as the x variable and distance as the y variable...

Please give Ace some feedback