Question

Google the first 50 numbers of the Fibonacci sequence (starting with 1) to answer the following questions:

         (a) Test to see if the leading digits conform to Benford’s law. Do this both graphically and analytically.

         (b) Using the first 10 odd numbers in the sequence as sample 1 and the first 10 even numbers in the sequence as sample 2, use Wilcoxon’s Rank-Sum to test the claim that the numbers come from different populations.

         (c) Repeat (b) using a t - Test.

Answer 1

Fibonacci sequence is a sequence of numbers in which each m]number (Fibonacci number) is the sum of the two preceding numbers.

First 25 numbers of the Fibonacci sequence are:

1,1,2,3,5,8,13,21,34,55,89,144,233,377,610,987,1597,2584,4181,6765,1094617711,28657,46368,75025.

r-code for creating Fibonacci series is given by,

x=c(1,1) ## Required series
length(x)=50 ##Required length
for(i in 3:50)
{
x[i]=x[i-1]+x[i-2]
}
x

b)

Here our hypothesis is

H0: Numbers comes from the same population Vs

H1: Numbers comes from different population.

r-code and output:

> sam1=c(1,1,3,5,13,21,55,89,233,377)

> sam2=c(2,8,34,144,610,2584,10946,46368,196418,832040)

> wilcox.test(sam1,sam2)

Wilcoxon rank sum test with continuity correction

data: sam1 and sam2

W = 20, p-value = 0.02569

alternative hypothesis: true location shift is not equal to 0

Here p-value is 0.0256 which is less that 0.05.(at 5% L.O.S.). Hence we reject H0.Which implies that numbers comes from the different population.

c)

r-code and output:

> t.test(sam1,sam2)

Welch Two Sample t-test

data: sam1 and sam2

t = -1.3171, df = 9, p-value = 0.2203

alternative hypothesis: true difference in means is not equal to 0

95 percent confidence interval:

-295761.81 78090.61

sample estimates:

mean of x mean of y

79.8 108915.4

By using t-test, p-value is 0.22 which is geater that 0.05. Hence at 5% level of significance, we accept H0. Which means numbers come from the same population.