Math Problem Statement
-
The data in the following table were obtained in a study of a student's score on a test and his/her grade point average. The study included eight randomly selected students. Find r, the correlation coefficient.
Test Score, X 98 105 100. 100. 106. 95. 116. 112
GPA, Y 2.1. 2.2. 3.0. 2.7. 2.4. 2.4. 3.4. 3.4
Group of answer choices
0.69
0.9
0.75
0.65
Solution
To find the correlation coefficient , we will use the formula for Pearson's correlation coefficient:
[ r = \frac{n \sum{xy} - (\sum{x})(\sum{y})}{\sqrt{[n\sum{x^2} - (\sum{x})^2][n\sum{y^2} - (\sum{y})^2]}} ]
Where:
- represents the test scores.
- represents the GPA.
- is the number of data points (in this case, ).
Step-by-step Calculation:
We are given the following data:
Test Score | GPA |
---|---|
98 | 2.1 |
105 | 2.2 |
100 | 3.0 |
100 | 2.7 |
106 | 2.4 |
95 | 2.4 |
116 | 3.4 |
112 | 3.4 |
First, we calculate the necessary sums:
Let’s calculate these sums step by step.
Calculations for the necessary sums:
-
-
-
So,
Now, we can substitute these values into the formula for :
[ r = \frac{8(2264.4) - (732)(21.6)}{\sqrt{[8(87790) - (732)^2][8(60.18) - (21.6)^2]}} ]
Let’s break this down:
Substituting these into the formula:
Thus, the correlation coefficient is approximately 0.75.
Final Answer:
0.75
Would you like further details or clarifications on any part of the calculation? Here are some follow-up questions to deepen the understanding:
- What does the value of the correlation coefficient indicate about the relationship between the test score and GPA?
- How does the Pearson correlation coefficient relate to linear regression analysis?
- What is the range of values for the correlation coefficient, and what do they represent?
- How can you interpret a correlation coefficient value near 1 or -1?
- How might the sample size affect the correlation coefficient's reliability?
Tip: The correlation coefficient is most useful when determining the strength and direction of a linear relationship between two variables. Always check the data visually (like using a scatter plot) to ensure linearity before using it!
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Math Problem Analysis
Mathematical Concepts
Statistics
Correlation Coefficient
Pearson's Correlation
Formulas
Pearson's correlation coefficient formula: r = (n Σxy - (Σx)(Σy)) / √[(n Σx² - (Σx)²)(n Σy² - (Σy)²)]
Theorems
Pearson's correlation theorem
Suitable Grade Level
Grades 10-12
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