I’m not sure what you’re trying to prove here. I don’t think it’s fair to compare a correlation coefficient, which gives you a single parameter and can be used without knowledge of the shapes of the underlying distributions, to a confidence interval for X around Z, which gives you 2 parameters, in a situation where the data actually is normally distributed. Furthermore, you are comparing a correlation coefficient of 0.6 to a measurement where Z-X is within 10% of a standard deviation of X! That’s outrageously accurate. For instance, the standard deviation of height of men is 3 inches; so when you want to know someone’s height X, you are given a measurement Z that is within 5/16″ of X. That’s almost within the range of measurement error you would get measuring X directly.
Make a comparison where you’re given correlations of .67 to 2 independent variables, versus a measurement that gives you 90% confidence of being within 2 standard deviations of the value of X, where Z is represented as being normally distributed around X, but is (unknown to Salviati) highly-skewed around X. I haven’t actually worked out the math to see what a fair test would be, but the example written up here is egregiously unfair.
I’m not sure what you’re trying to prove here. I don’t think it’s fair to compare a correlation coefficient, which gives you a single parameter and can be used without knowledge of the shapes of the underlying distributions, to a confidence interval for X around Z, which gives you 2 parameters, in a situation where the data actually is normally distributed. Furthermore, you are comparing a correlation coefficient of 0.6 to a measurement where Z-X is within 10% of a standard deviation of X! That’s outrageously accurate. For instance, the standard deviation of height of men is 3 inches; so when you want to know someone’s height X, you are given a measurement Z that is within 5/16″ of X. That’s almost within the range of measurement error you would get measuring X directly.
Make a comparison where you’re given correlations of .67 to 2 independent variables, versus a measurement that gives you 90% confidence of being within 2 standard deviations of the value of X, where Z is represented as being normally distributed around X, but is (unknown to Salviati) highly-skewed around X. I haven’t actually worked out the math to see what a fair test would be, but the example written up here is egregiously unfair.