The image shows a correlation hypothesis test involving the relationship between lemon imports (in metric tons) and car crash fatality rates (per 100,000 population). The task is to determine whether there is sufficient evidence to conclude a linear correlation between these two variables using a significance level of $\alpha = 0.05$.

Given Data:

• Lemon Imports: 232, 265, 357, 480, 530 (in metric tons)
• Crash Fatality Rate: 15.8, 15.7, 15.4, 15.2, 14.9 (per 100,000 population)

Steps to Conduct the Hypothesis Test:

1. Null Hypothesis ($H_0$): There is no linear correlation, i.e., $\rho = 0$.
2. Alternative Hypothesis ($H_1$): There is a linear correlation, i.e., $\rho \neq 0$.

To test for correlation, we:

1. Calculate the linear correlation coefficient (r).
2. Use the test statistic formula for correlation: $t = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}}$ where $n$ is the number of data points.
3. Calculate the P-value using the t-distribution with $n - 2$ degrees of freedom.
4. Compare the P-value with $\alpha = 0.05$ to draw a conclusion.

Answer Section: The possible true statements are:

1. $H_0: \rho = 0$ – This is always true for the null hypothesis in a correlation test.
2. The linear correlation coefficient $r = -0.985$.
3. The test statistic $t = -6.410$.
4. The P-value $= 0.002$.
5. Since the P-value $< 0.05$, we reject $H_0$ and conclude that there is sufficient evidence of a linear correlation between lemon imports and crash fatality rates.

Based on this, the correct statements from the answer section are:

1. $H_0: \rho = 0$
2. The linear correlation coefficient is $-0.985$.
3. The test statistic is $-6.410$.
4. The P-value is $0.002$.
5. The P-value is less than $0.05$, so there is sufficient evidence of a linear correlation.

Would you like more details on any step, or help with further calculations?

Related Questions:

1. How do you calculate the linear correlation coefficient?
2. What is the significance of the P-value in hypothesis testing?
3. What are the steps for interpreting a correlation hypothesis test result?
4. How do you construct a scatterplot to visualize the relationship between two variables?
5. How does changing the significance level $\alpha$ affect the conclusions of a hypothesis test?

Tip:

Always check if the sample size is large enough to make the correlation test statistically valid, as smaller samples can sometimes lead to misleading results.