You wish to determine if there is a linear correlation between the two variables at a significance level of
α

To address this hypothesis test for correlation, we need to calculate the correlation coefficient, $r$, for the data set and then compare it to the critical value $r_{c.v.} = 0.875$ at a significance level of $\alpha = 0.01$. We will then determine if there is sufficient evidence to conclude a statistically significant correlation between the variables.

Step 1: Calculate the Correlation Coefficient $r$

The formula for the sample correlation coefficient $r$ is:

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

where:

• $n$ is the number of data points,
• $\sum x$ is the sum of all $x$-values,
• $\sum y$ is the sum of all $y$-values,
• $\sum xy$ is the sum of the products of each $x$ and $y$ value,
• $\sum x^2$ is the sum of each $x$-value squared,
• $\sum y^2$ is the sum of each $y$-value squared.

Let’s calculate each of these values:

1. $n = 7$
2. $\sum x = 12.7 + 68.1 + 33.2 + 17.1 + 46.4 + 83.1 + 76.5$
3. $\sum y = 82.7 - 0.6 + 62.7 + 58.4 + 38.2 - 1.5 + 14.1$
4. $\sum xy$: Calculate each product $x \times y$ and then sum them.
5. $\sum x^2$: Square each $x$-value and sum.
6. $\sum y^2$: Square each $y$-value and sum.

Step 2: Interpret $r$

Once $r$ is calculated, we compare it to $r_{c.v.}$:

• If $|r| \geq 0.875$, there is sufficient evidence of a statistically significant correlation.
• If $|r| < 0.875$, there is insufficient evidence of correlation.

Calculation of $r$The calculated correlation coefficient $r$ is approximately $-0.953$.

Step 3: Conclusion

Since $|r| = 0.953$ is greater than the critical value $r_{c.v.} = 0.875$, we have sufficient evidence to support the claim that there is a statistically significant correlation between the two variables at the $\alpha = 0.01$ significance level.

Final Conclusion

There is sufficient sample evidence to support the claim that there is a statistically significant correlation between the two variables.

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Would you like further details on any step or have any questions?

Here are five related questions:

1. How does the significance level $\alpha$ affect the critical value in a correlation test?
2. What is the interpretation of a negative correlation coefficient like $r = -0.953$?
3. How does sample size impact the correlation coefficient’s reliability?
4. What alternative methods exist to assess correlation other than the Pearson correlation?
5. How does outlier presence affect the calculation and interpretation of correlation?

Tip: In hypothesis testing for correlation, a larger sample size generally makes it easier to detect small correlations as significant.