To determine whether there is a linear relationship between the year ($x$) and the annual high temperature ($y$), we conduct a hypothesis test for the slope of the regression line ($\beta_1$):

Step 1: Formulate the Hypotheses

• Null Hypothesis ($H_0$): $\beta_1 = 0$ (No linear relationship between year and temperature).
• Alternative Hypothesis ($H_1$): $\beta_1 \neq 0$ (There is a linear relationship).

Step 2: Significance Level

$\alpha = 0.01$

Step 3: Perform Linear Regression

We calculate the regression line parameters and conduct a hypothesis test using the t-statistic for the slope. Key values needed are:

1. Slope ($b_1$)
2. Intercept ($b_0$)
3. Standard Error of the Slope ($SE_{b_1}$)
4. t-Statistic: $t = \frac{b_1}{SE_{b_1}}$
5. p-value for the t-statistic.

I will calculate these values.import numpy as np import pandas as pd from scipy.stats import linregress

Data

x = np.array([3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]) y = np.array([14.72, 15.70, 12.98, 13.16, 14.04, 12.52, 14.50, 13.88, 12.66, 9.84, 10.52, 9.00])

Perform linear regression

slope, intercept, r_value, p_value, std_err = linregress(x, y)

Calculate t-statistic

t_stat = slope / std_err

Degrees of freedom

df = len(x) - 2

Organize results into a summary

results = { "Slope (b1)": slope, "Intercept (b0)": intercept, "Standard Error of Slope (SE_b1)": std_err, "t-Statistic": t_stat, "p-Value": p_value, "Degrees of Freedom": df, "R-Squared": r_value**2, } import ace_tools as tools; tools.display_dataframe_to_user(name="Linear Regression Results Summary", dataframe=pd.DataFrame([results])) Slope (b1) Intercept (b0) Standard Error of Slope (SE_b1) t-Statistic
0 -0.463357 16.731865 0.104607 -4.429513

p-Value  Degrees of Freedom  R-Squared  

0 0.001275 10 0.662397 Slope (b1) Intercept (b0) Standard Error of Slope (SE_b1) t-Statistic
0 -0.463357 16.731865 0.104607 -4.429513

p-Value  Degrees of Freedom  R-Squared  

0 0.001275 10 0.662397 successI have displayed the detailed results of the linear regression for your review. This includes the slope, intercept, standard error of the slope, t-statistic, p-value, degrees of freedom, and the R-squared value.

Step 4: Decision and Conclusion

1. Compare p-value to $\alpha$:

   • p-value = 0.001 (rounded), which is less than $\alpha = 0.01$.

2. Conclusion:

   • Reject the null hypothesis ($H_0$).
   • There is sufficient evidence to conclude that there is a linear relationship between the year and high temperature.

Would you like a deeper analysis or further questions? Here are some related topics to explore:

1. How is the R-squared value interpreted in the context of this analysis?
2. What does the negative slope suggest about temperature trends over time?
3. How would the conclusion change if $\alpha = 0.05$ instead of 0.01?
4. Can we make predictions using this regression line? How reliable are they?
5. How do outliers potentially affect the results of the regression?

Tip: Always assess the residuals of a regression model to ensure assumptions of linear regression are met.