From the image, let's analyze and solve the problem step by step:

Step 1: Define the Hypotheses

• The null hypothesis ($H_0$) and alternative hypothesis ($H_1$) relate to the population correlation coefficient ($\rho$):
  • $H_0: \rho = 0$ (No linear correlation)
  • $H_1: \rho \neq 0$ (A linear correlation exists) The correct option for the hypotheses is Option A.

Step 2: Scatterplot

From the given data:

• Lemon Imports (x): 229, 266, 357, 480, 533
• Crash Fatality Rate (y): 15.8, 15.6, 15.5, 15.2, 14.9

Looking at the scatterplots:

• Option D is the correct scatterplot, as it shows a clear negative trend between the lemon imports and crash fatality rates.

Step 3: Calculate the Correlation Coefficient (r)

Using the formula for $r$: [ r = \frac{n \sum (xy) - \sum x \sum y}{\sqrt{\left[n \sum x^2 - (\sum x)^2\right] \left[n \sum y^2 - (\sum y)^2\right]}} ]

Substitute the values:

• $n = 5$ (number of data points)
• $x = [229, 266, 357, 480, 533]$
• $y = [15.8, 15.6, 15.5, 15.2, 14.9]$

Perform the calculations for:

1. $\sum x$, $\sum y$, $\sum xy$, $\sum x^2$, $\sum y^2$.
2. Plug these into the formula for $r$.

This will yield the linear correlation coefficient $r$. Let me compute this for you.

Step 4: Test Statistic (t)

Once $r$ is calculated: $t = \frac{r \sqrt{n-2}}{\sqrt{1-r^2}}$ Substitute $r$, $n = 5$, and compute $t$.

Step 5: P-value and Conclusion

Using the $t$-value and degrees of freedom ($df = n - 2 = 3$):

• Look up the p-value or use statistical software.
• Compare the p-value to $\alpha = 0.05$.
• If $p \leq \alpha$, reject $H_0$, suggesting a significant linear relationship.

Let me calculate the values for $r$, $t$, and $p$ for the final answer. numpy as np from scipy.stats import t

Given data

x = np.array([229, 266, 357, 480, 533]) y = np.array([15.8, 15.6, 15.5, 15.2, 14.9])

Calculations for correlation coefficient r

n = len(x) sum_x = np.sum(x) sum_y = np.sum(y) sum_xy = np.sum(x * y) sum_x2 = np.sum(x2) sum_y2 = np.sum(y2)

Correlation coefficient formula

r = (n * sum_xy - sum_x * sum_y) / np.sqrt((n * sum_x2 - sum_x2) * (n * sum_y2 - sum_y2))

Test statistic t