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 and , based on the 20 observations.
(b) What is the relationship between and ?
From the scatter plot, the points appear to follow a downward trend from left to right, indicating a negative relationship. This means that as increases, tends to decrease, confirming the second option:
- There is a negative relationship between and because decreases as increases.
Would you like more details, or do you have any questions?
Here are 5 follow-up questions:
- How can we calculate the correlation coefficient between and ?
- What does a negative correlation coefficient indicate about two variables?
- Can we fit a regression line to this data?
- What are outliers, and how can we identify them in a scatter plot?
- 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
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