To solve this problem, I'll calculate the correlation coefficient $r$, perform the hypothesis test using the $t$-statistic, and compute the $p$-value. Let me extract the data and proceed with the calculations.

Given Data:

X (Year):
4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18

Y (Temperature in $^\circ C$):
22.7, 19.3, 18.4, 18.6, 19.9, 16.9, 15.8, 16.9, 16.8, 16.1, 14.6, 15.1, 10.6, 13.4, 12.3

I will now compute the correlation coefficient $r$. numpy as np import scipy.stats as stats import pandas as pd

Data

x = np.array([4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]) y = np.array([22.7, 19.3, 18.4, 18.6, 19.9, 16.9, 15.8, 16.9, 16.8, 16.1, 14.6, 15.1, 10.6, 13.4, 12.3])

Correlation coefficient calculation

r, _ = stats.pearsonr(x, y)

Hypothesis test

n = len(x) t_statistic = r * np.sqrt((n - 2) / (1 - r**2)) p_value = 2 * stats.t.sf(np.abs(t_statistic), df=n-2)

Results

{ "Correlation Coefficient (r)": round(r, 3), "t-Statistic": round(t_statistic, 3), "p-Value": round(p_value, 4) }Here are the results:

1. Correlation Coefficient ($r$): $-0.916$ (rounded to three decimal places)
2. t-Statistic: $-8.206$ (rounded to three decimal places)
3. p-Value: $0.0000$ (rounded to four decimal places)

Interpretation:

• The correlation coefficient ($r = -0.916$) indicates a strong negative relationship between the year and the annual high temperature.
• With a $p$-value of $0.0000$ (significant at $\alpha = 0.01$), we reject the null hypothesis $H_0$ and conclude there is a significant relationship between the year and the high temperature.

Would you like further details or explanations?

Related Questions:

1. How is the correlation coefficient interpreted in terms of strength and direction?
2. What does a negative $r$ value imply about the relationship between two variables?
3. Why do we use a $t$-test to check the significance of the correlation coefficient?
4. How does the sample size affect the reliability of the correlation coefficient?
5. What alternative methods can be used to measure the relationship between variables?

Tip: Always check the scatter plot of data to visually assess the relationship before relying on $r$ alone.