Let's tackle this step by step.

---

(a) Correlation Coefficient

The formula for the Pearson correlation coefficient $r$ is:

$r = \frac{\sum \left( x_i - \bar{x} \right) \left( y_i - \bar{y} \right)}{\sqrt{\sum \left( x_i - \bar{x} \right)^2 \sum \left( y_i - \bar{y} \right)^2}}$

First, I'll calculate $\bar{x}$, $\bar{y}$, and all necessary terms for the dataset to determine $r$. numpy as np

Data

x = np.array([39.2, 45.2, 41.6, 38.5, 2.5, 19.5, 53.8, 56.9, 59.2, 61.1]) y = np.array([104.5, 18.1, -36, -4.8, 270.2, 177.9, -6.3, -71.1, -50.1, 23])

Means of x and y

mean_x = np.mean(x) mean_y = np.mean(y)

Calculating the terms for r

numerator = np.sum((x - mean_x) * (y - mean_y)) denominator = np.sqrt(np.sum((x - mean_x)**2) * np.sum((y - mean_y)**2))

Correlation coefficient

r = numerator / denominator r**(a)** The correlation coefficient ($r$) for the data set is approximately:

$r = -0.8929$

---

(b) P-value for the correlation coefficient

To calculate the p-value, we use a t-distribution with degrees of freedom ($df = n - 2$):

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

Here, $n = 10$. I'll calculate $t$ and determine the p-value. scipy.stats import t

Sample size and degrees of freedom

n = len(x) df = n - 2

Calculating t statistic

t_stat = r * np.sqrt((n - 2) / (1 - r**2))

Calculating two-tailed p-value

p_value = 2 * t.sf(np.abs(t_stat), df) t_stat, p_value**(b)** The t-statistic is approximately:

$t = -5.6076$

The p-value for this correlation coefficient is:

$p\text{-value} = 0.0005$

---

(c) Conclusion about the correlation

Since the p-value $0.0005$ is much smaller than the significance level $\alpha = 0.10$, we reject the null hypothesis.

Conclusion: We have statistically significant evidence that there is a strong negative linear correlation between $x$ and $y$ in the population.

---

Do you want a detailed breakdown of these calculations? Here are some questions to consider next:

1. How does the significance level ($\alpha$) affect the conclusion?
2. Can we graph the data to visualize the correlation?
3. What does a negative correlation coefficient mean in practical terms?
4. How would the conclusion change with a higher p-value?
5. Is there a way to calculate $r$ using software or a calculator for verification?

Tip: Always verify statistical conclusions by checking the assumptions of the test (e.g., linearity, independence).