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

We made the assumption that the accident rate is Poisson distributed, so that the probability of k accident in a year is...

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

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

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

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

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

$\displaystyle \chi$ $\displaystyle \sigma$

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