Math Problem Statement

We are testing for a linear correlation between lemon imports and crash fatality rates. What are the hypotheses, the correct scatterplot, and the interpretation of the results?

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, ρ\rho, is different from 0.

Thus, the null and alternative hypotheses are:

  • H0:ρ=0H_0: \rho = 0 (no correlation)
  • H1:ρ0H_1: \rho \neq 0 (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 rr:

The correlation coefficient, rr, measures the strength of the linear relationship. Using the given data:

Lemon Imports (xx)Crash Fatality Rate (yy)
22915.9
26415.6
35815.5
48215.3
53014.8

[ 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 rr and the sample size (n=5n = 5), we compute the test statistic tt as:

t=rn21r2t = \frac{r \sqrt{n-2}}{\sqrt{1 - r^2}}

The P-value is then found based on the t-distribution with n2=3n-2 = 3 degrees of freedom.

c. Conclusion:

Compare the P-value to α=0.05\alpha = 0.05:

  • If P0.05P \leq 0.05, reject H0H_0.
  • If P>0.05P > 0.05, fail to reject H0H_0.

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 rr, tt, and PP-value, or explain further.

Here are five related questions you might explore:

  1. What does a correlation coefficient of 0 indicate in real-world terms?
  2. How does increasing the sample size affect the reliability of rr?
  3. What assumptions must be met for linear correlation analysis?
  4. Can you identify potential lurking variables that might explain this correlation?
  5. 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