### Problem 5 HW3 RStudio a) b) c) d) Researchers wondered whether the size of a person's brain was related to the individual's mental capacity. They selected a sample of right-handed introductory psychology students who had SAT scores higher than 1350. The subjects took the Wechsler Adult Intelligent Scale-Revisited to obtain their IQ scores. Magnetic Resonance Imaging (MRI) scans were performed at the same facility for the subjects. The scans consisted of 18 horizontal MRI images. The computer counted all pixels with nonzero gray scale in each of the 18 images, and the total count served as an index for the brain size (Source: Willerman L., Schultz R., Rutledge J.N., Bigler E., In Vivo Brain size and Intelligence, Intelligence (1991), 15: 223 - 228).

| Gender | MRI Count | IQ | |---------:|:-----------:|:------:| | Female | 816,932 | 133 | | Female | 951,545 | 137 | | Female | 991,305 | 138 | | Female | 833,868 | 132 | | Female | 856,472 | 140 | | Female | 852,244 | 132 | | Female | 790,619 | 135 | | Female | 866,662 | 130 | | Female | 857,782 | 133 | | Female | 948,066 | 133 | | Male | 949,395 | 140 | | Male | 1,001,121 | 140 | | Male | 1,038,437 | 139 | | Male | 965,353 | 133 | | Male | 955,466 | 133 | | Male | 1,079,490 | 141 | | Male | 924,059 | 135 | | Male | 955,003 | 139 | | Male | 935,494 | 141 | | Male | 949,589 | 144 |

a) Draw a scatter diagram treating MRI counts as the explanatory variable and IQ as the response variable. Comment on what you see.

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b) Would you say the correlation is low, moderate, or strong? positive or negative?

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c) A lurking variable in the analysis is gender. Draw a scatter diagram treating MRI count as the explanatory variable and IQ as the response variable, but use a different plotting symbol for each gender. What do you notice?

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d) Compute the linear correlation coefficient between MRI count and IQ for both females and males. Are MRI count and IQ linearly related? What is the moral?

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e) Find the least-squares regression lines for both females and males. What do you notice about the values of the slopes for both genders? What is the moral?

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