For a data set of brain volumes 

​(cm^3​) 

and IQ scores of 

sixsix 

​males, the linear correlation coefficient is 

requals=0.2040.204. 

Use the table available below to find the critical values of r. Based on a comparison of the linear correlation coefficient r and the critical​ values, what do you conclude about a linear​ correlation?

To determine whether the linear correlation coefficient \( r = 0.204 \) indicates a statistically significant linear relationship between brain volumes and IQ scores, we need to compare it to the critical values of \( r \) from a correlation table at a given significance level (e.g., 0.05).

### Step-by-Step Analysis:

1. **Determine the Degrees of Freedom**:  
   For correlation, the degrees of freedom (df) is calculated as:
   \[
   \text{df} = n - 2
   \]
   where \( n \) is the sample size. In this case, since we have data for 6 males:
   \[
   \text{df} = 6 - 2 = 4
   \]

2. **Find the Critical Value from the Table**:  
   Let's assume a typical significance level of 0.05 (5%) for a two-tailed test. For df = 4 and a significance level of 0.05, the critical value of \( r \) can be found in a standard critical values table for the Pearson correlation coefficient.

   From the critical values table for \( r \), at \( \alpha = 0.05 \) and df = 4, the critical value is approximately:

   \[
   r_{\text{critical}} \approx \pm 0.811
   \]

3. **Compare the Linear Correlation Coefficient to the Critical Value**:  
   Now, compare the absolute value of the linear correlation coefficient \( r = 0.204 \) to the critical value of \( \pm 0.811 \).

   Since:
   \[
   |r| = 0.204 \quad \text{is less than} \quad 0.811,
   \]
   \( r = 0.204 \) does **not exceed** the critical value.

4. **Conclusion**:  
   Because \( |r| = 0.204 \) is **less than** the critical value of \( 0.811 \), we **fail to reject** the null hypothesis that there is no linear correlation. This means:

   - There is **not enough evidence** to conclude that there is a significant linear correlation between brain volumes and IQ scores in this sample of six males.
   - The observed correlation could likely be due to random chance rather than a meaningful linear relationship.

Would you like more details or have any questions?

### Relative Questions to Expand Understanding:

1. What happens to the critical value if the sample size increases?
2. How do you interpret the linear correlation coefficient \( r \) if it is negative?
3. What are the possible reasons for finding a low correlation in this sample?
4. How would the conclusion change if the correlation coefficient \( r \) were 0.9?
5. What does a critical value tell us in the context of hypothesis testing for correlation?

### Tip:

When comparing the correlation coefficient to the critical value, always use the absolute value of \( r \) and consider the appropriate degrees of freedom and significance level.

Explore the linear correlation coefficient and critical values in statistics with a detailed example. Analyze whether a correlation between brain volumes and IQ scores is statistically significant based on given data. Suitable for advanced high school students and college-level learners.

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