In Exercises 5–8, use the runs test with a significance level of α = 0.05. (All data are listed in order by row.)


Odd and Even Digits in Pi A New York Times article about the calculation of decimal places of π noted that “mathematicians are pretty sure that the digits of π are indistinguishable from any random sequence.” Given below are the first 25 decimal places of π. 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?

Verified step by step guidance

1. Step 1: Identify the sequence of odd (O) and even (E) digits from the given 25 decimal places of π. Odd digits are 1, 3, 5, 7, 9, and even digits are 0, 2, 4, 6, 8. Convert the sequence into a string of O's and E's based on this classification.
2. Step 2: Count the total number of runs in the sequence. A run is defined as a consecutive sequence of the same type (either all O's or all E's). For example, 'OOEEO' has three runs: 'OO', 'EE', and 'O'.
3. Step 3: Calculate the expected number of runs and the variance of the number of runs under the null hypothesis of randomness. Use the formulas: \( E(R) = \frac{2n_1n_2}{n_1 + n_2} + 1 \) and \( \text{Var}(R) = \frac{2n_1n_2(2n_1n_2 - n_1 - n_2)}{(n_1 + n_2)^2(n_1 + n_2 - 1)} \), where \( n_1 \) is the number of odd digits and \( n_2 \) is the number of even digits.
4. Step 4: Compute the test statistic \( Z \) using the formula \( Z = \frac{R - E(R)}{\sqrt{\text{Var}(R)}} \), where \( R \) is the observed number of runs, \( E(R) \) is the expected number of runs, and \( \text{Var}(R) \) is the variance.
5. Step 5: Compare the computed \( Z \)-value to the critical values for a two-tailed test at \( \alpha = 0.05 \). If \( |Z| \) is greater than the critical value, reject the null hypothesis of randomness. Otherwise, fail to reject the null hypothesis and conclude that the sequence appears random.

Key Concepts

Runs Test

The runs test is a non-parametric statistical test used to determine the randomness of a sequence. It analyzes the occurrence of runs, which are sequences of similar items, to assess whether the observed pattern deviates from what would be expected in a random sequence. In this context, it will help evaluate if the distribution of odd and even digits in the decimal places of π is random.

Significance Level (α)

The significance level, denoted as α, is the threshold for determining whether a statistical result is significant. In hypothesis testing, a common significance level is 0.05, which indicates a 5% risk of concluding that a difference exists when there is none. In this exercise, if the p-value from the runs test is less than 0.05, it suggests that the sequence of digits is not random.

Randomness in Sequences

Randomness in sequences refers to the lack of predictable patterns or trends in the arrangement of elements. In the context of the digits of π, the claim is that they should appear indistinguishable from a random sequence. Testing for randomness involves statistical methods, like the runs test, to verify if the distribution of odd and even digits aligns with what would be expected in a truly random sequence.