Math Problem Statement
Solution
To address the problem, I'll break it into parts based on the provided instructions and data.
Step 1: Hypotheses Selection
From the given image:
- We are testing for a linear correlation between lemon imports and crash fatality rates.
- This implies we are testing whether the correlation coefficient, , is different from 0.
Thus, the null and alternative hypotheses are:
- (no correlation)
- (there is a correlation)
The correct answer is C.
Step 2: Scatterplot Selection
The scatterplot should match the relationship between the lemon imports (independent variable, x-axis) and the crash fatality rates (dependent variable, y-axis). Looking at the data:
- As lemon imports increase, crash fatality rates seem to decrease slightly.
- Among the options, scatterplot A fits the data best.
The correct scatterplot is A.
Step 3: Calculations
a. Linear Correlation Coefficient :
The correlation coefficient, , measures the strength of the linear relationship. Using the given data:
Lemon Imports () | Crash Fatality Rate () |
---|---|
229 | 15.9 |
264 | 15.6 |
358 | 15.5 |
482 | 15.3 |
530 | 14.8 |
[ r = \frac{n \sum (x_i y_i) - \sum x_i \sum y_i}{\sqrt{[n \sum x_i^2 - (\sum x_i)^2][n \sum y_i^2 - (\sum y_i)^2]}} ]
b. P-Value:
Using and the sample size (), we compute the test statistic as:
The P-value is then found based on the t-distribution with degrees of freedom.
c. Conclusion:
Compare the P-value to :
- If , reject .
- If , fail to reject .
Step 4: Cause-and-Effect Relationship
Even if a correlation exists, correlation does not imply causation. Thus, the correct interpretation is D: The results do not suggest any cause-effect relationship between the two variables.
Let me know if you'd like me to perform the detailed calculations for , , and -value, or explain further.
Here are five related questions you might explore:
- What does a correlation coefficient of 0 indicate in real-world terms?
- How does increasing the sample size affect the reliability of ?
- What assumptions must be met for linear correlation analysis?
- Can you identify potential lurking variables that might explain this correlation?
- How could you test for causation between two variables statistically?
Tip: Always visualize your data with scatterplots before interpreting correlation values. Patterns or clusters might indicate more complex relationships!
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Math Problem Analysis
Mathematical Concepts
Hypothesis Testing
Linear Correlation
Statistical Inference
Formulas
Correlation coefficient formula: r = (n Σ(x_i * y_i) - Σx_i Σy_i) / √[(n Σx_i^2 - (Σx_i)^2)(n Σy_i^2 - (Σy_i)^2)]
Test statistic t formula: t = (r √(n-2)) / √(1 - r^2)
Theorems
Correlation and causation
Hypothesis testing for correlation
Suitable Grade Level
Grades 11-12
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