Question

Odd and Even Digits in Pi A New York Times article about the calculation of decimal places of\(\pi \)noted that “mathematicians are pretty sure that the digits of\(\pi \)are indistinguishable from any random sequence.” Given below are the first 25 decimal places of\(\pi \). Test for randomness in the way that odd (O) and even (E) digits occur in the sequence. Based on the result, does the statement from the New York Times appear to be accurate?

1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3

Short answer

There is enough evidence to conclude that the sequence of the decimal places of\(\pi \)is random.

Thus, the mathematicians have accurately stated that the digits of \(\pi \) are indistinguishable from any random sequence.

Step-by-step solution

Given information

A sequence is given showing the decimal places of \(\pi \).

Frame the statistical hypothesis and the decision rule

The mathematicians have stated that the digits of\(\pi \)are indistinguishable from any random sequence.

This means that the given sequence of decimal places of\(\pi \)is a random sequence.

The null hypothesis is as follows:

The given sequence of decimal places of\(\pi \)is random.

The alternative hypothesis is as follows:

The given sequence of decimal places of\(\pi \)is not random.

If the value of the number of runs (G) is less than or equal to a smaller critical value or greater than or equal to a larger critical value, the null hypothesis is rejected.

Arrange the data

Represent the odd digits of the decimal places with O.

Represent the even digits of the decimal places with E.

The table below shows the indicated symbols:

Decimal Places	Symbol
1	O
4	E
1	O
5	O
9	O
2	E
6	E
5	O
3	O
5	O
8	E
9	O
7	O
9	O
3	O
2	E
3	O
8	E
4	E
6	E
2	E
6	E
4	E
3	O
3	O

Step 4:Calculate the test statistic

The sequence is as follows:

O

E

O

O

O

E

E

O

O

O

E

O

O

O

O

E

O

E

E

E

E

E

E

O

O

Now, the number of times O occurs is denoted by\({n_1}\), and the number of times E occurs is denoted by\({n_2}\).

Thus,

\(\begin{array}{l}{n_1} = 14\\{n_2} = 11\end{array}\)

The runs of the sequence are formed as follows:

\(\underbrace O_{{1^{st}}run}\underbrace E_{{2^{nd}}run}\underbrace {OOO}_{{3^{rd}}run}\underbrace {EE}_{{4^{th}}run}\underbrace {OOO}_{{5^{th}}run}\underbrace E_{{6^{th}}run}\underbrace {OOOO}_{{7^{th}}run}\underbrace E_{{8^{th}}run}\underbrace O_{{9^{th}}run}\underbrace {EEEEEE}_{{{10}^{th}}run}\underbrace {OO}_{{{11}^{th}}run}\)

The number of runs denoted byG is equal to 11.

Here,\({n_1} \le 20\)and\({n_2} \le 20\).

Thus, the test statistic is G, and the level of significance\(\left( \alpha \right)\)is equal to 0.05.

The critical values of G for\({n_1} = 14\)and\({n_2} = 11\)are 8 and 19.

The value of G equal to 11 is neither less than or equal to 8 nor greater than or equal to 19. Thus, the decision is fail to reject the null hypothesis.

Step 5:Conclusion of the test

There is not enough evidence to conclude that the given sample is not random.

The sequence of decimal places of\(\pi \)is random.

Thus, the statement of the mathematicians that the digits of\(\pi \)are indistinguishable from any random sequence is accurate.