Math Problem Statement

Listed below are annual data for various years. The data are weights (metric tons) of imported lemons and car crash fatality rates per 100,000 population. Construct a scatterplot, find the value of the linear correlation coefficient r, and find the P-value using α = 0.05. Is there sufficient evidence to conclude that there is a linear correlation between lemon imports and crash fatality rates? Do the results suggest that imported lemons cause car fatalities?

Solution

The task requires calculating the missing test statistic tt, interpreting the results, and determining the correct answer based on the given data.

Solution Steps:

  1. Identify the correlation coefficient rr: Given r=0.471r = -0.471.

  2. Calculate the test statistic tt: The formula for tt is: t=rn21r2t = \frac{r \sqrt{n - 2}}{\sqrt{1 - r^2}} Where nn is the sample size (number of data points).

  3. Determine the sample size: From the data, n=5n = 5 (5 data points).

  4. Substitute into the formula: t=0.471521(0.471)2t = \frac{-0.471 \sqrt{5 - 2}}{\sqrt{1 - (-0.471)^2}}

    Simplify step-by-step:

    • r2=(0.471)2=0.221841r^2 = (-0.471)^2 = 0.221841
    • 1r2=10.221841=0.7781591 - r^2 = 1 - 0.221841 = 0.778159
    • n2=31.732\sqrt{n - 2} = \sqrt{3} \approx 1.732
    • 1r2=0.7781590.882\sqrt{1 - r^2} = \sqrt{0.778159} \approx 0.882
    • t=0.4711.7320.882t = \frac{-0.471 \cdot 1.732}{0.882}

    Final computation: t0.8167320.8820.926t \approx \frac{-0.816732}{0.882} \approx -0.926

    So, t0.926t \approx -0.926.

  5. Interpret the P-value and significance level:

    • Given P=0.669P = 0.669, which is greater than α=0.05\alpha = 0.05, we fail to reject the null hypothesis.
  6. Conclusion:

    • The correlation is not statistically significant.
    • There is insufficient evidence to suggest a linear relationship between lemon imports and crash fatality rates.
  7. Correct Answer:

    • The correct option is A: The results do not suggest any cause-effect relationship between the two variables.

Would you like me to explain any of the steps in more detail or clarify further? Here are some follow-up questions you might explore:

  1. What does a P-value signify in hypothesis testing?
  2. How is the test statistic tt used to assess significance?
  3. Why can correlation not imply causation?
  4. How does the sample size nn impact the test statistic?
  5. Can a negative rr still indicate a strong linear relationship?

Tip: Always examine whether the context and data make a causal relationship plausible before making conclusions!

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Math Problem Analysis

Mathematical Concepts

Linear correlation
Hypothesis testing
Statistical significance

Formulas

Correlation coefficient formula
Test statistic formula: t = r√(n-2) / √(1-r²)

Theorems

Null hypothesis testing
Significance level and P-value comparison

Suitable Grade Level

Grades 11-12 or introductory college statistics