Questions: Researchers wondered whether the size of a person's brain was related to the individual's mental capacity. They selected a sample of 3 females and 3 males and measured their MRI counts and IQ scores. The data is reported on the right. MRI IQ MRI IQ 856,473 141 1,079,550 135 857,781 135 1,001,121 141 991,305 138 965,355 132 Treat the MRI count as the explanatory variable. Compute the linear correlation coefficient between MRI count and IQ for both the males and the females. Do you believe that MRI count and IQ are linearly related? The linear correlation coefficient for females is . The linear correlation coefficient for males is 7. (Round to three decimal places as needed.) Are MRI count and IQ linearly related? The linear correlation coefficient for females is close to so linear relation exists between MRI count and IQ for females. The linear correlation coefficient for males is close to so linear relation exists

Solution Steps

To find the linear correlation coefficient \( r \) for females, we use the formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Where:

• \( \text{Cov}(X,Y) = -1962.0 \)
• \( \sigma_X = 77470.465 \)
• \( \sigma_Y = 3.0 \)

Substituting the values:

\[ r = \frac{-1962.0}{77470.465 \times 3.0} = -0.008 \]

Thus, the linear correlation coefficient for females is:

\[ \text{Linear correlation coefficient for females: } -0.008 \]

Next, we calculate the linear correlation coefficient \( r \) for males using the same formula:

\[ r = \frac{\text{Cov}(X,Y)}{\sigma_X \sigma_Y} \]

Where:

• \( \text{Cov}(X,Y) = 32317.5 \)
• \( \sigma_X = 58410.634 \)
• \( \sigma_Y = 4.583 \)

Substituting the values:

\[ r = \frac{32317.5}{58410.634 \times 4.583} = 0.121 \]

Thus, the linear correlation coefficient for males is:

\[ \text{Linear correlation coefficient for males: } 0.121 \]

For females, since \( r \approx -0.008 \), we conclude that there is no strong linear relationship between MRI count and IQ.

For males, since \( r \approx 0.121 \), we also conclude that there is no strong linear relationship between MRI count and IQ.

Final Answer