To solve this problem, we need to address several steps systematically:

1. Scatterplot Construction: The data provided involves shoe print lengths (cm) and heights (cm). Based on the scatterplot choices, the points plotted must align with the given data. By inspecting the scatterplots provided, the correct choice appears to be B, where the data points match the given values.

2. Linear Correlation Coefficient (r): The correlation coefficient, $r$, measures the strength and direction of the linear relationship between two variables. To compute $r$: $r = \frac{\sum \left( (x - \bar{x})(y - \bar{y}) \right)}{\sqrt{\sum (x - \bar{x})^2 \sum (y - \bar{y})^2}}$ Here, $x$ represents the shoe print length and $y$ represents height. We will calculate this using the data provided.

3. P-Value and Significance Test: Once $r$ is calculated, we determine whether the linear correlation is significant using the t-test: $t = r \sqrt{\frac{n - 2}{1 - r^2}}$ The p-value is obtained using the degrees of freedom $df = n - 2$, where $n$ is the number of data pairs.

4. Analysis and Conclusion: If the p-value is less than the significance level ($\alpha = 0.05$), we conclude that there is sufficient evidence to support the claim that shoe print length can be used to estimate height.

Let me calculate these values for you from scipy.stats import t

Data from the table

shoe_print = np.array([30.5, 30.5, 30.4, 30.7, 26.9]) height = np.array([176.9, 184.2, 181.9, 179.2, 177.3])

Calculate means

mean_shoe_print = np.mean(shoe_print) mean_height = np.mean(height)

Calculate the numerator and denominator for r

numerator = np.sum((shoe_print - mean_shoe_print) * (height - mean_height)) denominator = np.sqrt(np.sum((shoe_print - mean_shoe_print)**2) * np.sum((height - mean_height)**2)) r = numerator / denominator

Degrees of freedom

n = len(shoe_print) df = n - 2

Calculate t-statistic

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

Calculate p-value (two-tailed)

p_value = 2 * t.sf(abs(t_statistic), df)

r, pThe linear correlation coefficient $r$ is approximately 0.446, indicating a moderate positive correlation between shoe print length and height.

The p-value is approximately 0.452, which is greater than the significance level ($\alpha = 0.05$).

Conclusion:

There is insufficient evidence to support the claim of a significant linear correlation between shoe print length and height at the 0.05 significance level.

Would you like further details or explanations? Here are some related questions for further exploration:

1. How is the correlation coefficient $r$ computed step-by-step?
2. What does a p-value greater than $\alpha$ indicate in hypothesis testing?
3. Could using a larger dataset improve the reliability of this analysis?
4. How does the t-distribution play a role in calculating the p-value?
5. What other statistical methods could we use to estimate height from shoe print data?

Tip: Ensure the sample size is sufficiently large to