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From the question below, I require a regression model for the first part. To get the coefficients of regression, I used the correlation provided and the standard deviations to compute the co-variance.I then computed the slope by dividing the Co-variance with the variance of the independent variable. To compute the constant of the model, I used the mean of the dependent variable the slope and mean of the independent variable. My main problem is the second part. Part b. I wonder if I require the coefficient of determination which can easily be computed from correlation but doesn't seem right to me or what other coefficient or way to compute it.

A group of married couples takes an IQ test. The average husband's IQ is $105$ with an SD of $15$ and the average wife's IQ is $110$ with an SD of $10$. The correlation between husband's and wife's IQ is $0.5.$

a) A man has an IQ of $75$, what would you predict his wife's IQ is? (I understand how to do this part)

b) Of all men with an IQ of 75, about what percent are smarter than their wives

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For a regression model of $w$ (wife) on $h$ (husband), we have $$w|h=\langle{w}\rangle|h+\epsilon$$ i.e. $w$ given $h$ is an expected value (mean $\langle{w}\rangle|h$) plus a residual ($\epsilon$).

The "regression equation" gives the expected value. Under the standard linear regression (OLS) assumptions the noise $\epsilon$ is normally distributed with zero-mean.

So you have two parts:

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  • $\begingroup$ I appreciate your help GeoMatt22. Looking at your work has confused me a bit. After computing my model, I got simple linear regression model of W=75+0.333X. However what I see you suggest is a conditional regression. Is it any different from my model?? $\endgroup$
    – Ben
    Commented May 6, 2017 at 3:34
  • $\begingroup$ The full regression model would formally be $w=ah+b+\sigma_0\epsilon$ where the final term is random error (here $\epsilon$ is now a standard normal). As it is zero-mean, it averages out in $\langle{w}|h\rangle$, i.e. averaging over $w$'s while keeping $h$ fixed. Technically $\sigma_0$ is a third coefficient that you estimate (that is my first bullet). FYI an alternative formulation of the $a$ and $b$ coefficients is that the ratio of $z$-scores is equal to the correlation. $\endgroup$
    – GeoMatt22
    Commented May 6, 2017 at 3:44

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