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I am trying to model relationship between length of stay of patients in hospital(Y) vs Age in years(X). The data set I've got doesn't specify the unit of length of stay. So now estimated value of my coefficient for age is $ b_1 = 0.084 $. So if a patient was 12 years older than he is currently then his stay will increase "on an average" by 1 unit of time. However I can only give a precise (say 99% precise) estimate by confidence interval. But this confidence interval will have small boundaries like 0.003 to 0.3, which is slightly unexplanabale for layman listener.

One way is that I create a new random variable $ b_2 =12b_1$ and its standard error will be $ 12 \cdot \hat s\{b_0\} $. And this variable will be t-distributed with $ n-2 $ degrees of freedom. Using this I can build 99% confidence interval on the true value of my new random variable $b_2$ . And then I can use it for interpretation.

Am I doing it correctly?

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Yes, if you want to convert from stay in years to stay in months, it looks like you're doing it correctly - both the mean and standard error scale by the same factor and the t-ratio is unchanged.

(You didn't actually specify what the units were, before or after the conversion - so I had to guess. If they were some other unit, then your use of 12 might be incorrect.)

In some situations it might be better still to convert to days.

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