An energy economist studying the growth and decay of both the coal and natural gas industries wanted to leverage data collected by environmental scientists. Particularly, the economist wanted to study the link between the total yearly carbon emissions from both energy sources. They looked at  
30
  years of total yearly carbon emissions for a particular nation (measured in million metric tons of carbon dioxide) for both forms of energy. The data is provided below. Use Rguroo to calculate the correlation coefficient  
r
  between the two variables. Round your answer to two decimal places.
MMTons CO2 Natural Gas	MMTons CO2 from Coal
144.975	1552.524
173.578	1519.755
168.448	1517.232
183.205	1537.343
178.596	1528.600
190.391	1658.477
212.140	1708.757
222.428	1626.367
210.728	1810.590
219.543	1733.430
257.485	1773.156
256.113	1828.702
274.249	1991.943
292.051	1807.748
316.249	1849.436
272.586	1948.157
310.474	1874.673
313.879	1923.322
340.916	1999.025
356.621	2057.241
368.335	1959.583
362.115	1805.112
399.512	1809.525
392.326	1745.279
468.730	1570.326
434.550	1513.523
456.864	1512.334
503.702	1369.385
561.474	1245.502
490.599	1258.338

Provide your answer below:

$$

To calculate the correlation coefficient $ r $ between the total yearly carbon emissions from natural gas and coal, we'll use the Pearson correlation formula. The Pearson correlation coefficient formula is:

\[
r = \frac{n(\sum xy) - (\sum x)(\sum y)}{\sqrt{[n\sum x^2 - (\sum x)^2][n\sum y^2 - (\sum y)^2]}}
\]

Where:
- $ n $ is the number of data points,
- $ x $ represents the carbon emissions from natural gas,
- $ y $ represents the carbon emissions from coal.

### Steps to calculate:
1. Sum the products of paired values $ xy $.
2. Sum the individual values of $ x $ (natural gas emissions) and $ y $ (coal emissions).
3. Sum the squares of $ x $ and $ y $.
4. Plug all the values into the correlation formula.

Instead of going step-by-step manually, I'll run the calculations and give you the correlation coefficient rounded to two decimal places.

The Pearson correlation coefficient $ r $ between the total yearly carbon emissions from natural gas and coal is approximately $ -0.26 $.

This suggests a weak negative correlation between the two variables, meaning that as carbon emissions from natural gas increase, emissions from coal tend to slightly decrease, but the relationship is not strong.

Would you like any further explanation on this or have questions? Here are 5 related questions:

1. What does a negative correlation coefficient indicate in environmental data?
2. How does Pearson's correlation differ from Spearman's correlation?
3. How can we interpret weak correlations in energy sector studies?
4. What other statistical analyses can be performed on this dataset?
5. How might policy changes affect the correlation between natural gas and coal emissions?

**Tip:** When correlation is low, it suggests other factors or variables may influence the relationship, and further analysis might be needed.

An energy economist studying the growth and decay of both the coal and natural gas industries wanted to leverage data collected by environmental scientists. Particularly, the economist wanted to study the link between the total yearly carbon emissions from both energy sources. They looked at 30 years of total yearly carbon emissions for a particular nation (measured in million metric tons of carbon dioxide) for both forms of energy. Use Rguroo to calculate the correlation coefficient r between the two variables. Round your answer to two decimal places.

This problem involves calculating the Pearson correlation coefficient between the total yearly carbon emissions from natural gas and coal over 30 years in a specific nation. The data is analyzed to determine the relationship between emissions from these two energy sources, providing insight into environmental trends. Suitable for undergraduate students studying statistics or environmental science.

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