Inter-arrival times of births for exponential distribution.

gistfile1.txt

# statinf-exp-distro

While I was searching for real examples of exponential distribution, I came across this interesting piece[1].

"To see an example of a distribution that is approximately exponential, we will look at the interarrival time of babies. 
On December 18, 1997, 44 babies were born in a hospital in Brisbane, Australia. The times of birth for all 44 babies were 
reported in the local paper; you can download the data from http://thinkstats.com/babyboom.dat. "

Later, I came across the Centers for Disease Control and Prevention web site[2] that has data (~200 MB !) on births in the US. 
The User Guide PDF [3] document has the columns for date and time of birth. I want to use this data in a way similar to course
project of 'Statistical Inference' course. The project description as below:

    In this project you will investigate the exponential distribution in R and compare it with the Central Limit Theorem. 
    The exponential distribution can be simulated in R with rexp(n, λ ) where λ  is the rate parameter. 
    The mean of exponential distribution is 1/λ  and the standard deviation is also 1/λ . Set λ = 0.2 for all 
    of the simulations. You will investigate the distribution of averages of 40 exponentials. 
    
    Note that you will need to do a thousand simulations. 
    
    Illustrate via simulation and associated explanatory text the properties of the distribution of the mean of 40 exponentials.  
    You should 
    1. Show the sample mean and compare it to the theoretical mean of the distribution. 
    2. Show how variable the sample is (via variance) and compare it to the theoretical variance of the distribution.
    3. Show that the distribution is approximately normal.

Thus, with the births dataset, to prove the convergence, do the following steps sound valid?

    Establish that the data has an exponential distribution. (How do I find the λ value?)
    Take a random sample (size = 10) of the birth inter-arrival time. Calculate its mean. 
    Repeat this many times (n=1000). Plot it. 
    Repeat for size=50, size=100, size=250 and size=500.
    
    Then, all the plots (for sizes 50, 100 and 500) should converge around normal distribution for 1000 simulations. 
    
Further, some hypothesis tests could also be formulated.
    
1 - http://greenteapress.com/thinkstats/html/thinkstats005.html 
2 - http://www.cdc.gov/nchs/data_access/Vitalstatsonline.htm
3 - ftp://ftp.cdc.gov/pub/Health_Statistics/NCHS/Dataset_Documentation/DVS/natality/UserGuide2013.pdf