91Ó°ÊÓ

Consider the following data set: $$ \begin{array}{lllllllll} \hline x & 5 & 6 & 7 & 7 & 8 & 8 & 8 & 8 \\\ \hline y & 4.2 & 5 & 5.2 & 5.9 & 6 & 6.2 & 6.1 & 6.9 \\ \hline x & 9 & 9 & 10 & 10 & 11 & 11 & 12 & 12 \\ \hline y & 7.2 & 8 & 8.3 & 7.4 & 8.4 & 7.8 & 8.5 & 9.5 \\ \hline \end{array} $$ (a) Draw a scatter diagram with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20 . (b) Compute the linear correlation coefficient. (c) Now multiply both \(x\) and \(y\) by 2 . (d) Draw a scatter diagram of the new data with the \(x\) -axis starting at 0 and ending at 30 and with the \(y\) -axis starting at 0 and ending at 20. Compare the scatter diagrams. (e) Compute the linear correlation coefficient. (f) Conclude that multiplying each value in the data set by a nonzero constant does not affect the correlation between the variables.

Step by step solution

01

Draw the original scatter diagram

Plot the points \((x, y)\) on a graph with the x-axis ranging from 0 to 30 and the y-axis ranging from 0 to 20. The points are: (5, 4.2), (6, 5), (7, 5.2), (7, 5.9), (8, 6), (8, 6.2), (8, 6.1), (8, 6.9), (9, 7.2), (9, 8), (10, 8.3), (10, 7.4), (11, 8.4), (11, 7.8), (12, 8.5), (12, 9.5).

02

Compute the linear correlation coefficient

Use the formula for Pearson’s correlation coefficient: \( r = \frac{n(\text{Σ}xy) - (\text{Σ}x)(\text{Σ}y)}{\root( \root( {\text{Σ}x^2 - (\text{Σ}x)^2 / n)}}{\text{Σ}y^2 - (\text{Σ}y)^2 / n)}\), where \ is the number of data pairs. Calculate \Σx, Σy, Σxy, Σx^2\, and \Σy^2\, then substitute into the formula to find \r\.

03

Multiply x and y by 2

Multiply each value in the original \(x\) and \(y\) dataset by 2. The new values are: \(x: [10, 12, 14, 14, 16, 16, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24]\) and \(y: [8.4, 10, 10.4, 11.8, 12, 12.4, 12.2, 13.8, 14.4, 16, 16.6, 14.8, 16.8, 15.6, 17, 19]\).

04

Draw the scatter diagram with multiplied values

Plot the newly multiplied points \((2x, 2y)\) on a graph with the x-axis ranging from 0 to 30 and the y-axis ranging from 0 to 20. The points are: (10, 8.4), (12, 10), (14, 10.4), (14, 11.8), (16, 12), (16, 12.4), (16, 12.2), (16, 13.8), (18, 14.4), (18, 16), (20, 16.6), (20, 14.8), (22, 16.8), (22, 15.6), (24, 17), (24, 19). Compare this diagram to the first scatter diagram.

05

Compute the new linear correlation coefficient

Using the same formula for Pearson's correlation coefficient, calculate the value for the new data points \(2x\) and \(2y\). Note that the linear correlation coefficient \r\ remains unchanged.

06

Conclude the effect of multiplication by a nonzero constant

Since the correlation coefficient obtained in Steps 2 and 5 are the same, it shows that multiplying each value in the dataset by a nonzero constant does not affect the correlation coefficient. The linear relationship between the variables remains consistent.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

scatter diagram

A scatter diagram, or scatter plot, is a visual representation of the relationship between two variables. Each point on the plot represents a data pair from your dataset, with the horizontal axis (x-axis) mapping one variable and the vertical axis (y-axis) mapping the other.
To create a scatter diagram:

• Draw two perpendicular lines to represent the x and y axes, and label them.
• Mark your data points on the graph by plotting the (x,y) pairs.

This exercise involved plotting the points: (5, 4.2), (6, 5), (7, 5.2), (7, 5.9), (8, 6), (8, 6.2), (8, 6.1), (8, 6.9), (9, 7.2), (9, 8), (10, 8.3), (10, 7.4), (11, 8.4), (11, 7.8), (12, 8.5), (12, 9.5).
After plotting these points, you should observe the pattern they form on the graph. If the points appear to follow a certain direction, it suggests a correlation between x and y.

Pearson’s correlation coefficient

Pearson's correlation coefficient, denoted as \(r\), is a statistic that measures the strength and direction of a linear relationship between two variables. The formula for Pearson’s correlation coefficient is:
\( r = \frac{n(\text{Σ}xy) - (\text{Σ}x)(\text{Σ}y)}{\root( \root( {\text{Σ}x^2 - (\text{Σ}x)^2 / n)}}{\text{Σ}y^2 - (\text{Σ}y)^2 / n)}\)
where:

• \(\text{Σ}xy\) is the sum of the product of paired scores.
• \(\text{Σ}x\) and \(\text{Σ}y\) are the sums of x and y scores respectively.
• \(\text{Σ}x^2\) and \(\text{Σ}y^2\) are the sums of squared x and y scores respectively.
• \(n\) is the number of data pairs.

In the given exercise, you first calculate this using the original datasets. By plugging in the required sums into the formula, you arrive at a value for \(r\).
An \(r\) value close to 1 or -1 indicates a strong linear relationship, while a value close to 0 indicates a weak linear relationship.

data transformation

Data transformation involves changing the scale or units of the data while maintaining the intrinsic relationship between the variables. In this exercise, the transformation was achieved by multiplying each value of \(x\) and \(y\) by 2.
The transformed dataset included new values:
\(x: [10, 12, 14, 14, 16, 16, 16, 16, 18, 18, 20, 20, 22, 22, 24, 24]\)
and
\(y: [8.4, 10, 10.4, 11.8, 12, 12.4, 12.2, 13.8, 14.4, 16, 16.6, 14.8, 16.8, 15.6, 17, 19]\).
Plotting these transformed points on a scatter diagram helps visually compare with the original dataset. Despite the transformation, the linear correlation coefficient remains the same, confirming that such scaling does not affect the relationship between variables.
In conclusion, data transformation serves many purposes, such as simplifying the dataset, but it does not alter the fundamental correlations.