Math Problem Statement

The following 20 observations are for two quantitative variables, x and y.

Observationxy

1

−21

24

2

−32

49

3

2

10

4

29

−16

5

−13

12

6

23

−26

7

−12

29

8

−22

37

9

15

−3

10

3

−3   Observationxy

11

−36

48

12

36

−27

13

11

−18

14

−32

33

15

22

−14

16

−3

16

17

−15

20

18

12

19

19

−19

−9

20

−7

−20

(a)

Develop a scatter diagram for the relationship between x and y.

A scatter diagram has 20 points.

The horizontal axis is labeled x and ranges from −40 to 40.

The vertical axis is labeled y and ranges from −40 to 60.

The points plotted from x values of −36 to 36 are evenly scattered between y values of −27 to 49.   A scatter diagram has 20 points.

The horizontal axis is labeled x and ranges from −40 to 40.

The vertical axis is labeled y and ranges from −40 to 60.

The points are plotted diagonally from top left to bottom right. The points are somewhat scattered.   A scatter diagram has 20 points.

The horizontal axis is labeled x and ranges from −40 to 40.

The vertical axis is labeled y and ranges from −40 to 60.

The points are plotted such that for each point the value of the x coordinate is same as the value of the y coordinate.   A scatter diagram has 20 points.

The horizontal axis is labeled x and ranges from −40 to 40.

The vertical axis is labeled y and ranges from −40 to 60.

The points are plotted diagonally from bottom left to top right. The points are somewhat scattered.

(b)

What is the relationship, if any, between x and y?

There is a positive relationship between x and y because y increases as x increases.There is a negative relationship between x and y because y decreases as x increases.    There is no relationship between x and y.There is a positive relationship between x and y because y decreases as x increases.There is a negative relationship between x and y because y increases as x increases.

Solution

The scatter diagram above shows the relationship between the variables xx and yy, based on the 20 observations.

(b) What is the relationship between xx and yy?

From the scatter plot, the points appear to follow a downward trend from left to right, indicating a negative relationship. This means that as xx increases, yy tends to decrease, confirming the second option:

  • There is a negative relationship between xx and yy because yy decreases as xx increases.

Would you like more details, or do you have any questions?

Here are 5 follow-up questions:

  1. How can we calculate the correlation coefficient between xx and yy?
  2. What does a negative correlation coefficient indicate about two variables?
  3. Can we fit a regression line to this data?
  4. What are outliers, and how can we identify them in a scatter plot?
  5. How does scaling of axes impact the interpretation of a scatter plot?

Tip: Always look at the scatter plot to visually assess the relationship before calculating correlation coefficients!

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Math Problem Analysis

Mathematical Concepts

Statistics
Scatter Plot
Correlation

Formulas

Correlation coefficient formula
Linear regression equation

Theorems

Pearson's correlation theorem

Suitable Grade Level

Grades 10-12