To determine if there is a significant correlation between $x$ and $y$ at the 0.05 significance level, we need to compute the correlation coefficient ($r$) and test its significance. Here's the step-by-step process:

Steps:

1. Compute the Pearson correlation coefficient ($r$): [ r = \frac{n\sum(xy) - \sum x \sum y}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}} ]

2. Calculate the degrees of freedom (df): $df = n - 2$ where $n$ is the number of data points.

3. Find the critical value of $r$ at $\alpha = 0.05$ for the given $df$.

4. Compare the computed $r$ to the critical value. If $|r|$ exceeds the critical value, the correlation is significant.

Let me compute these values for your data.import numpy as np import pandas as pd from scipy.stats import pearsonr, t

Input data

x = np.array([4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]) y = np.array([18.58, 18.1, 21.22, 27.74, 27.96, 28.28, 23.4, 35.12, 26.84, 36.96, 41.38])

Number of data points

n = len(x)

Compute Pearson correlation coefficient

r, p_value = pearsonr(x, y)

Degrees of freedom

df = n - 2

Calculate critical value of r for a two-tailed test at alpha=0.05

alpha = 0.05 t_critical = t.ppf(1 - alpha/2, df)

Output results

correlation_results = { "Pearson r": r, "p-value": p_value, "Degrees of freedom": df, "Significant at 0.05": p_value < alpha }

correlation_results{'Pearson r': 0.8685610977238304, 'p-value': 0.0005222409060658683, 'Degrees of freedom': 9, 'Significant at 0.05': True}{'Pearson r': 0.8685610977238304, 'p-value': 0.0005222409060658683, 'Degrees of freedom': 9, 'Significant at 0.05': True}success### Results:

1. Pearson Correlation Coefficient ($r$): $0.8686$
2. p-value: $0.00052$
3. Degrees of Freedom (df): $9$
4. Significance: Since the p-value ($0.00052$) is less than the significance level ($0.05$), the correlation is significant.

Conclusion:

The data shows a significant correlation between $x$ and $y$ at the 0.05 significance level.

Would you like further details or visualizations?

Related Questions:

1. How is the critical value of $r$ calculated for a given significance level?
2. What does the strength and direction of the correlation coefficient imply?
3. Can we visualize the data with a scatterplot and a trendline?
4. What are the assumptions of Pearson's correlation test?
5. How would the significance change if we used a 0.01 significance level?

Tip:

Always verify the assumptions of normality and linearity when using Pearson's correlation to ensure valid results.