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Do the successive digits in the decimal expansion of \(\pi\) behave as though they were selected from a random number table (or came from a computer's random number generator)? a. Let \(p_{0}\) denote the long-run proportion of digits in the expansion that equal 0 , and define \(p_{1}, \ldots\), \(p_{9}\) analogously. What hypotheses about these proportions should be tested, and what is df for the chi-squared test? b. \(H_{0}\) of part (a) would not be rejected for the nonrandom sequence \(012 \ldots 901 \ldots 901 \ldots\) Consider nonoverlapping groups of two digits, and let \(p_{i j}\) denote the long-run proportion of groups for which the first digit is \(i\) and the second digit is \(j\). What hypotheses about these proportions should be tested, and what is df for the chi-squared test? c. Consider nonoverlapping groups of 5 digits. Could a chi-squared test of appropriate hypotheses about the \(p_{i j k l m}\) 's be based on the first 100,000 digits? Explain. d. The paper "Are the Digits of \(\pi\) an Independent and Identically Distributed Sequence?" (Amer. Statist., 2000: 12-16) considered the first \(1,254,540\) digits of \(\pi\), and reported the following \(P\)-values for group sizes of \(1, \ldots, 5\) digits: \(.572, .078, .529, .691, .298\). What would you conclude?

Step by step solution

01

Define hypotheses for part (a)

In part (a), we need to test if each digit in the decimal expansion of \(\pi\) is equally likely. The null hypothesis \(H_0\) is that each digit occurs with equal probability, i.e., \(p_0 = p_1 = \ldots = p_9 = 0.1\). The alternative hypothesis \(H_a\) is that at least one proportion does not equal 0.1.

02

Determine degrees of freedom (df) for part (a)

For the chi-squared test, the degrees of freedom is calculated as the number of categories minus one. Since there are 10 digits (0 through 9), the degrees of freedom is \(10 - 1 = 9\).

03

Define hypotheses for part (b)

In part (b), we consider pairs of digits. The null hypothesis \(H_0\) is that each pair \((i, j)\) occurs with probability \(p_{ij} = 0.01\), assuming independence and equal likelihood. The alternative hypothesis \(H_a\) is that at least one pair probability \(p_{ij} eq 0.01\).

04

Determine degrees of freedom (df) for part (b)

When considering pairs of digits, there are 100 possible pairs from \(00\) to \(99\). The degrees of freedom for the chi-squared test is \(100 - 1 = 99\).

05

Consider groups of 5 digits for part (c)

In part (c), we analyze groups of 5 digits. The number of different 5-digit combinations is \(10^5 = 100,000\). However, since we only have 100,000 digits, which would yield only 20,000 groups (as each group takes 5 digits), chi-squared testing is infeasible due to insufficient data points compared to the number of combinations.

06

Interpret P-values in part (d)

In part (d), given that all the \(P\)-values are above typical significance levels (e.g., 0.05), none of the tests would lead to rejecting the null hypothesis. Thus, based on this data, the hypothesis that the digits behave randomly is not rejected.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Randomness of Pi digits

The digits of \(\pi\), an irrational number, continue infinitely without repeating. It has been a long-standing question whether these digits appear as a random sequence. To explore this, one can use the chi-squared test to analyze the distribution of these digits to see if they resemble a random distribution.

When we talk about randomness in the digits of \(\pi\), we suggest that each digit from 0 to 9 appears with equal probability in its decimal expansion. For example, in a perfectly random sequence, each digit should show up approximately 10% of the time. This is what we mean when we refer to the 'randomness' of \(\pi\)'s digits.

To check this, we set up a null hypothesis, \(H_0\), which states that each digit has an equal chance of appearing. The alternative hypothesis, \(H_a\), would be that at least one digit does not follow this pattern. By applying a chi-squared test, we can statistically determine if there is a significant deviation from this expected distribution.

Hypothesis testing

Hypothesis testing is a statistical method that enables us to make decisions or inferences about population parameters. It involves two opposing hypotheses: the null hypothesis (\(H_0\)) and the alternative hypothesis (\(H_a\)).

• The null hypothesis (\(H_0\)) is a statement of no effect or no difference, suggesting that any observed effect in the data is due to random variation.
• The alternative hypothesis (\(H_a\)) indicates that there is an actual effect, implying that the observed effect in the data is real and not random.

The chi-squared test is often used in hypothesis testing when we want to compare observed frequencies with expected frequencies of categorical data. The greater the discrepancy between observed and expected data, the less likely the null hypothesis is true.

In examining the digits of \(\pi\), a chi-squared test can be applied by using the frequencies of each digit and comparing it to what would be expected under the assumption of randomness. If the test leads to rejecting \(H_0\), there is evidence that the digits of \(\pi\) do not behave randomly. Otherwise, \(H_0\) is accepted, indicating randomness.

Degrees of freedom in statistics

Degrees of freedom (df) in statistics signify the number of independent values or quantities which can be assigned to a statistical distribution. They are a crucial part of setting up and interpreting tests such as the chi-squared test.

For the chi-squared test, degrees of freedom are calculated based on the number of categories in the data. Specifically, it's the number of categories minus 1. For example, when testing the randomness of individual digits of \(\pi\), there are 10 categories (digits 0 through 9), leading to 9 degrees of freedom (\(10 - 1 = 9\)).

This concept becomes particularly important when considering groups of digits. For pairs of digits, there are \(10 \times 10 = 100\) combinations, resulting in 99 degrees of freedom for the test. As the number of categories increases, so does the complexity of analyzing the data, especially if there are insufficient data points to cover every possible category combination.

Understanding degrees of freedom helps ensure that the statistical test is correctly set up, allowing researchers to draw meaningful conclusions about the randomness and independence of \(\pi\)’s digits.