Question 1163537
```python?code_reference&code_event_index=2
import numpy as np
import scipy.stats as stats

# Data
taxes = np.array([21860, 17463, 18232, 15691, 43998, 33674, 15233, 16760, 18247, 16777, 15164, 36087, 31021, 42137, 14374, 38935, 25336, 22947, 16177, 29226])
size = np.array([2486, 2447, 1851, 1000, 5688, 2547, 2221, 1913, 2013, 1368, 1393, 3099, 2808, 3322, 1582, 4096, 4047, 2471, 3551, 2845])

# n
n = len(taxes)

# r_xy
r, p_val = stats.pearsonr(size, taxes)

# t-statistic for correlation
# t = r * sqrt(n-2) / sqrt(1-r^2)
t_stat = r * np.sqrt(n-2) / np.sqrt(1 - r**2)

print(f"Sample size n: {n}")
print(f"Correlation coefficient r_xy: {r:.4f}")
print(f"Test statistic t: {t_stat:.3f}")
print(f"p-value: {p_val:.10f}")


```
```text?code_stdout&code_event_index=2
Sample size n: 20
Correlation coefficient r_xy: 0.7587
Test statistic t: 4.940
p-value: 0.0001056345

```
Based on the data provided for the 20 homes, here is the statistical analysis of the relationship between house size and property taxes:

### a-1. Sample Correlation Coefficient $r_{xy}$
Using the formula for the Pearson correlation coefficient:
$$r_{xy} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$$
The calculated value is:
**$r_{xy} = 0.7587$**

### a-2. Interpretation of $r_{xy}$
Since $0.7587$ is a positive value, it indicates that as the size of the house increases, the property taxes also tend to increase.
**Correct Choice: a. The correlation coefficient indicates a positive linear relationship.**

---

### b. Competing Hypotheses
To determine if the population correlation coefficient ($\rho_{xy}$) differs from zero (indicating a significant relationship), we set up a two-tailed test:
* **$H_0: \rho_{xy} = 0$** (No correlation)
* **$H_A: \rho_{xy} \neq 0$** (Correlation exists)

**Correct Choice: a.**

---

### c-1. Test Statistic
The test statistic for the correlation coefficient follows a $t$-distribution with $n - 2$ degrees of freedom ($df = 18$):
$$t = \frac{r\sqrt{n-2}}{\sqrt{1-r^2}} = \frac{0.7587\sqrt{18}}{\sqrt{1-0.7587^2}} \approx 4.940$$
**Value of the test statistic: $4.940$**

### c-2. $p$-value
For a $t$-statistic of $4.940$ with $18$ degrees of freedom, the $p$-value is approximately $0.0001$.
**Correct Choice: a. $p\text{-value} < 0.01$**

---

### d. Conclusion to the Test
At the **1% significance level** ($\alpha = 0.01$), we compare the $p$-value to $\alpha$:
* Since $0.0001 < 0.01$, we **reject the null hypothesis ($H_0$)**.
* There is sufficient evidence to conclude that a linear relationship exists between house size and property taxes.

**Correct Choice: a. Reject $H_0$; we can state size and property taxes are correlated.**