I am self-studying an applied statistics course (reading the curriculum, doing the recommended exercises, doing old projects etc.) and I am now trying to deal with this problem:
How significant is a drop from 4039 to 3611 traffic related injuries per year?
The process (of being injured due to traffic) is a Poisson process with a given rate.
With \(\lambda > 3000 \) the normal distribution with \(\mu=\lambda\) and \(\sigma^2 = \lambda\) is a good approximation.
To test if this is significant I do a "two sample test". My text book have a test called / described as: "two Gaussian samples with known variance: are the means the same?".
This means the null hypothesis is that the means are the same. This can be rejected (or accepted, of course) and hence the alternative (that they are not the same) made seem more probable.
Define a new random variable which is the difference in the two means and test how compatible this is with zero:
\(Y_\mathrm{inj} = X_{\mathrm{inj, 2012}} - X_{\mathrm{inj, 2011}}\)
The standard deviation of the new random variable:
\(\sigma_{Y_{inj}} = \sqrt{\sigma_{inj,2012}^2 + \sigma_{inj,2011}^2} = \sqrt{4039+3611} = 87.46\)
The the actual difference in terms of \(\sigma\)
\(\frac{Y_{inj}}{\sigma_{Y_{inj}}} = \frac{4039-3611}{87.46} = 4.89\)
So now I know that the actual differens (which according to the null-hypothesis should be 0) is \(4.89\sigma\) different from 0.
How do I turn this in to a statement like "This drop is significant at level \(\alpha\)". ?
I know, or think I know, that it has to do with this probability:
\(Pr(\mu - 4.89\sigma \leq x \leq \mu+4.89\sigma) = 0.9999989916\)
The amount of events lying within \(4.89\sigma\) from the mean (in both directions) on a Gaussian. But I don't know how to make the last final statement of the significance.
Any help would be appreciated.
Best regards
Jonas
How significant is a drop from 4039 to 3611 traffic related injuries per year?
The process (of being injured due to traffic) is a Poisson process with a given rate.
With \(\lambda > 3000 \) the normal distribution with \(\mu=\lambda\) and \(\sigma^2 = \lambda\) is a good approximation.
To test if this is significant I do a "two sample test". My text book have a test called / described as: "two Gaussian samples with known variance: are the means the same?".
This means the null hypothesis is that the means are the same. This can be rejected (or accepted, of course) and hence the alternative (that they are not the same) made seem more probable.
Define a new random variable which is the difference in the two means and test how compatible this is with zero:
\(Y_\mathrm{inj} = X_{\mathrm{inj, 2012}} - X_{\mathrm{inj, 2011}}\)
The standard deviation of the new random variable:
\(\sigma_{Y_{inj}} = \sqrt{\sigma_{inj,2012}^2 + \sigma_{inj,2011}^2} = \sqrt{4039+3611} = 87.46\)
The the actual difference in terms of \(\sigma\)
\(\frac{Y_{inj}}{\sigma_{Y_{inj}}} = \frac{4039-3611}{87.46} = 4.89\)
So now I know that the actual differens (which according to the null-hypothesis should be 0) is \(4.89\sigma\) different from 0.
How do I turn this in to a statement like "This drop is significant at level \(\alpha\)". ?
I know, or think I know, that it has to do with this probability:
\(Pr(\mu - 4.89\sigma \leq x \leq \mu+4.89\sigma) = 0.9999989916\)
The amount of events lying within \(4.89\sigma\) from the mean (in both directions) on a Gaussian. But I don't know how to make the last final statement of the significance.
Any help would be appreciated.
Best regards
Jonas
