91Ó°ÊÓ

A car is being driven at an average speed range of \(50-70 \mathrm{kmph}\). The table shows distances between selected cities and the time taken by the car to cover these kilometers. a. Calculate the correlation of the numbers shown in the part a table by using a computer or statistical calculator. $$ \begin{array}{|c|c|} \hline \text { Distance (km) } & \text { Time (hrs) } \\ \hline 120 & 2 \\ \hline 294 & 4 \\ \hline 160 & 3 \\ \hline 340 & 6 \\ \hline 310 & 5 \\ \hline \end{array} $$ b. The table for part b shows the same information, except that the distance was converted to meters by multiplying the number of kilometers by 1000 . What happens to the correlation when numbers are multiplied by a constant? $$ \begin{array}{|c|c|} \hline \text { Distance (m) } & \text { Time (hrs) } \\ \hline 120000 & 2 \\ \hline 294000 & 4 \\ \hline 160000 & 3 \\ \hline 340000 & 6 \\ \hline 310000 & 5 \\ \hline \end{array} $$ c. Suppose the \(0.5\) hour that is lost at toll booths is added to the hours during each travel, no matter how long the distance is. The table for part \(c\) shows the new data. What happens to the correlation when a constant is added to cach number? $$ \begin{array}{|c|c|} \hline \text { Distance (km) } & \text { Time (hrs) } \\ \hline 120 & 2.5 \\ \hline 294 & 4.5 \\ \hline 160 & 3.5 \\ \hline 340 & 6.5 \\ \hline 310 & 5.5 \\ \hline \end{array} $$

Step by step solution

01

- Calculation of Correlation for Part a

Firstly, we need to calculate the correlation for the given distances and times in part a. We put the distances as variable x and the times as variable y, then use our statistical calculator or software to compute the correlation coefficient.

02

- Transformation of Variables and Effects on Correlation for Part b

In the second table, each distance (x-value) is multiplied by 1000 to convert from kilometers to meters. However, this multiplication by a constant does not affect the correlation coefficient. It remains the same as in part (a). This is because correlation is a dimensionless quantity and only measures the strength of the linear relationship, which isn't changed by this transformation.

03

- Addition of Constant Time and Effects on Correlation for Part c

In part (c), a constant time of 0.5 hours is added to every recorded time. This translates every time value upward by the same amount, but does not affect the scatter or 'spread' of the time data relative to the distance data. Consequently, the correlation remains the same as in parts (a) and (b), because adding (or subtracting) a constant to either or both variables does not change the correlation.

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.

Transformation of Variables

In statistics, transforming variables involves modifying data using mathematical operations. For example, converting kilometers into meters by multiplying by 1000 is a transformation. Such operations usually aim to enhance analysis, clarify data, or meet certain assumptions for statistical models.
In the exercise, we transformed distances measured in kilometers into meters for part b by multiplying by 1000. However, this transformation, while changing the unit of measurement, did not affect existing relationships between variables like correlation.
A critical point to note is that correlation, as a measure of linear relationship strength, is unaffected by scaling or shifting. So, even if you transform variables, the underlying association measured by correlation remains unchanged.

Linear Relationship

A linear relationship refers to a direct proportional connection between two variables, where changes in one variable lead to proportional changes in another. Mathematically, this is represented by a straight line on a graph.
In the exercise provided, we analyze the linear relationship between distance and time, where it is expected that more distance results in more time, assuming constant speed. Such relationships are common in real-world situations where a linear increase in one variable results in a linear increase in the other.
The strength of this relationship can be quantified using the correlation coefficient, showing how closely data points fit a linear pattern. When analyzing linear relationships, it's essential to visualize the data using scatter plots to confirm if the trend truly appears linear.

Correlation Coefficient

The correlation coefficient is a numerical value that indicates the strength and direction of a linear relationship between two variables. It ranges from -1 to 1. A correlation of 1 means perfect positive linear relationship; -1 indicates perfect negative linear relationship; and 0 means no linear relationship.
In the exercise, calculating the correlation coefficient involves analyzing the relationship between distance traveled and time. Despite transformations or adding constants, as seen in parts b and c, the coefficient itself remains unaffected.
Key takeaways about correlation:

• It doesn't depend on the scale of data; multiplying or dividing by a constant won't change it.
• Adding or subtracting a constant to all data points doesn't affect it.
• It's useful for understanding and predicting relationships but not for defining causality.

Understanding correlation helps in grasping the basic patterns within data, especially when exploring linear relationships.