PIL import Image

Load the images to ensure they were uploaded correctly and to review their content

image_1_path = "/mnt/data/1733196024357mwwse5z9.jpg" image_2_path = "/mnt/data/17331960243574zg26tfb.jpg"

image_1 = Image.open(image_1_path) image_2 = Image.open(image_2_path)

image_1.show() image_2.show()I have opened the uploaded images. Based on the content in the images, let me help you calculate the required statistics (correlation coefficient, hypotheses, test statistic, and p-value) using the provided data. numpy as np from scipy.stats import linregress, t

Data from the image for President and Opponent Heights

president_heights = np.array([175, 178, 190, 179, 184, 181, 194, 187, 173, 186, 182, 186, 188]) opponent_heights = np.array([180, 176, 179, 174, 175, 184, 177, 183, 179, 175, 169, 192, 170])

Calculate the linear correlation coefficient

correlation_coefficient, p_value = linregress(president_heights, opponent_heights)[:2]

Null hypothesis H0: rho = 0 (no correlation)

Alternative hypothesis H1: rho != 0 (there is a correlation)

n = len(president_heights) t_statistic = correlation_coefficient * np.sqrt((n - 2) / (1 - correlation_coefficient**2)) p_value_two_tailed = 2 * (1 - t.cdf(abs(t_statistic), df=n - 2))

Rounding values for output

correlation_coefficient_rounded = round(correlation_coefficient, 3) t_statistic_rounded = round(t_statistic, 2) p_value_rounded = round(p_value_two_tailed, 3)

{ "Correlation Coefficient (r)": correlation_coefficient_rounded, "t-Statistic": t_statistic_rounded, "P-Value": p_value_rounded }Here are the calculated results:

1. Correlation Coefficient (r): 0.008
2. t-Statistic: 0.03
3. P-Value: 0.981

Conclusion:

• Since the p-value (0.981) is much greater than the significance level ($\alpha = 0.01$), we fail to reject the null hypothesis.
• There is insufficient evidence to support a claim of linear correlation between the heights of the presidents and their opponents.

Do you need further details or assistance? Here are five related questions to consider:

1. How is the linear correlation coefficient calculated?
2. What does a p-value represent in hypothesis testing?
3. Why do we use the t-statistic in this context?
4. Can the result differ if a larger dataset is used?
5. What is the implication of failing to reject the null hypothesis?

Tip: Always visualize the data with a scatterplot to better understand the relationship before statistical testing. Let me know if you’d like to analyze the scatterplot options in the image!