# Comparing means.

• Jan 7th 2012, 03:42 PM
question
Comparing means.
Hi there

Need some help with the below

Prior to the introduction of hard shoulder running on a stretch of the M1, the long term mean accident rate was 6 fatalities/year. In the first year of the hard shoulder running scheme 7 fatalities were observed at the same stretch of motorway. The authorities are concerned this suggests an increase in the underlying mean accident rate. Carry out a formal test to assess whether this is the case, clearly stating any assumptions made.

Thanks a lot for any help
• Jan 7th 2012, 06:28 PM
chisigma
Re: Comparing means.
We made the assumption that the accident rate is Poisson distributed, so that the probability of k accident in a year is...

$P(k,\lambda)= \frac{\lambda^{k}\ e^{- \lambda}}{k!}$ (1)

... where $\lambda$ is the expected value of accidents in a year. Before the introduction of the hard shoulder a value of $\lambda=6$ was obeserved. If $\lambda$ didn't change after the introduction of the hard shoulder, then the probability of 7 accidents is...

$P(7,6) \sim .138$ (2)

If after the introduction of the hard shoulder we have [for example...] $\lambda=7$, then the probability of 7 accidents is...

$P(7,7) \sim .149$ (3)

The difference between (2) and (3) is not significative, so that further statistical data are necessary in order to valuate the increasing if the accident rate...

Kind regards

$\chi$ $\sigma$
• Jan 7th 2012, 11:55 PM
CaptainBlack
Re: Comparing means.
Quote:

Originally Posted by question
Hi there

Need some help with the below

Prior to the introduction of hard shoulder running on a stretch of the M1, the long term mean accident rate was 6 fatalities/year. In the first year of the hard shoulder running scheme 7 fatalities were observed at the same stretch of motorway. The authorities are concerned this suggests an increase in the underlying mean accident rate. Carry out a formal test to assess whether this is the case, clearly stating any assumptions made.

Thanks a lot for any help

The null hypothesis is that there is no difference in which case your estimate of the mean number of accidents per year is 6.5, and the number has a Poisson distribution with this mean.

So your test is a one-sided test to see if the probability of 7 or more accidents in a year is "small" under the null-hypothesis.

CB