Density Plots by Year

These plots are not Normally Distributed (or ‘Bell Shaped’ which is the distribution most people know) but that is ok.

When dealing with probabilities involving time intervals we can use a Poission distribution which has a ‘skewed’ look like the plots above.

Plots 2-4 look to be distributed Poisson, but the first is questionable.

Amount Paid by Year

      2014   2015   2016   2017
Year 31276 167248 108620 116483

2014 looks extremely low with amount paid in 2014 being roughly $31,000, and the others years exceeding $100,000.

The 2014 numbers could be skewing our data so we can do a ‘z-test’ to see if the 2014 paid value is statistically different than the other years.

z-value:

round((31276-105906)/(56129.96/sqrt(4)),4)
[1] -2.6592

A z-value of -2.66 produces a p-value of 0.0034 which is significantly different than the other years at an alpha level of 0.01. This validates our theory that the 2014 year is skewing our data, and we can now remove it.

Now we can find out the probability of certain numbers of injuries per year. The average number of injuries per year is 45.33. Since we know it is distributed Poisson we can calculate probabilities.

round(ppois(57, lambda = 45.33,lower.tail = FALSE),4)
## [1] 0.0393
round(ppois(62, lambda = 45.33, lower.tail = FALSE),4)
## [1] 0.0075

In statistics we use bench marks of 95% and 99%.

95% is used most frequently but if you want to be extra careful you can use a 99% bench mark.

From the numbers above you can see that there is a 5% probability of GSI having 57 injuries or more in one year. Also there is a 1% probability of 62 or more injuries in a year. What this is telling us is these events are rare. Statistically we would not expect to see 57 or more injuries, and it would be extremely unlikely to see 62.

With these numbers we can now see how much GSI should expect to pay in a given year.

       2015   2016   2017 Total.Paid Average.Claim
Paid 167248 108620 116483     392351      2884.934

From these numbers we see that 2015-2017 have similar numbers. The Average Claim is valuable because now we can estimate how much GSI will pay in future years.

                            95%      99%
Projected total claims 164441.2 178865.9

It would be statistically rare to see more than 57 injuries in a year so it would be rare to see GSI pay $164,441.20 a year on injuries. Taking a more cautious approach we can use the 99% number.

It would be extremely rare to see 62 injuries in a year so it would be extremely rare to see GSI pay $178,865.90 a year on injuries.

Now lets look at an estimate for the average number of injuries per year.

[1] 2884.934

Now that we have the average amount GSI pays per injury we can forecast how much GSI will pay per year using a confidence interval

poisson.test(45, conf.level = 0.95)

    Exact Poisson test

data:  45 time base: 1
number of events = 45, time base = 1, p-value < 2.2e-16
alternative hypothesis: true event rate is not equal to 1
95 percent confidence interval:
 32.82331 60.21354
sample estimates:
event rate 
        45

This is telling us that 95% of the time the true mean number of injuries will fall between 33-60 injuries per year.

Now we can just take the average injury figure and multiply it by these number to get a range for how much GSI might pay per year.

2884.934*33
## [1] 95202.82
2884.934*60
## [1] 173096
2884.934*45
## [1] 129822

With 95% accuracy GSI can expect to have 33-60 injuries per year. This will cost between $95,202.82 - $173,096.00.

Finally the average number of injuries we would expect per year is 45, so we would expect on average to pay $129,822.