Question

Use the P-value Approach for all hypothesis tests. Assume that all samples are randomly obtained.

The first significant digit in any number is: 1, 2, 3, 4, 5, 6, 7, 8, and 9. Though we may think that each digit would appear with equal frequency, this is not true. Physicist, Frank Benford, discovered that in many situations where counts accumulate over time that the first digit in the count follows a particular pattern. The probabilities of occurrence to the first digit in a number, known as Benford’s Law, are shown below.

1st Digit	1	2	3	4	5	6	7	8	9
Probability (Benford’s Law)	.301	.176	.125	.097	.079	.067	.058	.051	.046

Bloomberg’s web site (www.bloomberg.com) gives information on the stock price and trading volume of the members of the S&P 500 stock index. Suppose a random sample of 100 stocks on a given day were selected from the site, and the first digit of the daily stock volume (total number of shares traded in a given day) was recorded below.

1st Digit	1	2	3	4	5	6	7	8	9
Probability (Benford’s Law)	.301	.176	.125	.097	.079	.067	.058	.051	.046
Stock Volume (frequency)	28	16	18	8	7	6	6	5	6

Using a 10% level of significance, test whether the first digits in the stock price follow the distribution of probabilities given by Benford’s Law. Be sure to verify the requirements for the test.

Answer 1

Null hypothesis:Ho: first digits in the stock price follow the distribution of probabilities given by Benford’s Law.

Alternate hypothesis:Ha: : first digits in the stock price does not follow the distribution of probabilities given by Benford’s Law.

degree of freedom =categories-1=

7

for 7 df and 0.1 level of signifcance critical region       χ2=

12.017

as expected frequency in 9 as 1st digit =np=100*0.046 =4.6 is less than 5 ; therefore we add this category with 8 as 1st digit ;

other requirements are met as expected frequency in each category is at least 5.

Applying chi square test:

	relative	observed	Expected	residual	Chi square
category	frequency	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.301	28	30.10	-0.38	0.147
2	0.176	16	17.60	-0.38	0.145
3	0.125	18	12.50	1.56	2.420
4	0.097	8	9.70	-0.55	0.298
5	0.079	7	7.90	-0.32	0.103
6	0.067	6	6.70	-0.27	0.073
7	0.058	6	5.80	0.08	0.007
8 or more	0.097	11	9.70	0.42	0.174
total	1.000	100	100		3.367

as test statistic 3.367 is not higher then critical value we can nt reject null hypothesis

we can conclude that first digits in the stock price follow the distribution of probabilities given by Benford’s Law.