Question

Daily Lottery Numbers Listed below are the daily numbers (daytime drawing) for the Pennsylvania State Lottery for February 2007. Using O for odd and E for even, test for randomness at \(\alpha=0.05\). $$\begin{array}{lllllll}270 & 054 & 373 & 204 & 908 & 121 & 121 \\ 804 & 116 & 467 & 357 & 926 & 626 & 247 \\\ 783 & 554 & 406 & 272 & 508 & 764 & 890 \\ 441 & 964 & 606 & 568 & 039 & 370 & 583\end{array}$$

Short answer

The sequence of odd and even numbers is random at \(\alpha=0.05\).

Step-by-step solution

Identify Even and Odd Numbers

List each lottery number and replace it with 'O' for odd and 'E' for even. You'll be looking at each individual digit of the lottery numbers: \[\begin{array}{ccc} EOE & OEE & OOO & EOE & EOE & OEO & OEO \ EOE & EEO & OOE & OOO & EEO & EOE & OEO \ OOO & EEO & EOE & EEO & EOE & EOE & EEO \ OEO & EEO & EOE & EOE & OOO & EOE & OOE \end{array}\]

Count O and E Sequences

Count the number of occurrences of 'O' and 'E'. For instance, with the sequence, "EOE, OEE", break it down by digit to count how many O's (odd) and E's (even) appear. Add these numbers for the entire data set.

Construct the Test Hypothesis

The null hypothesis (\(H_0\)) states that the sequence of odd and even numbers is random. The alternative hypothesis (\(H_a\)) suggests a non-random arrangement.

Determine the Test Statistic

Use a runs test for randomness. A "run" is a sequence of like items (e.g., E's or O's), and switching from E to O and vice versa starts a new run. Count these runs.

Calculate Expected Runs

Calculate the expected number of runs \(E(R)\) using the formula: \[ E(R) = \frac{2n_1n_2}{n_1 + n_2} + 1 \] where \(n_1\) is the number of E's and \(n_2\) is the number of O's.

Calculate the Standard Deviation of Runs

Compute the standard deviation of runs \(\sigma_R\) as: \[ \sigma_R = \sqrt{ \frac{2n_1n_2(2n_1n_2 - n_1 - n_2)}{(n_1 + n_2)^2(n_1 + n_2 - 1)}} \]

Apply Runs Test for Randomness

Calculate the Z-value as:\[ Z = \frac{R - E(R)}{\sigma_R} \] Where \(R\) is the actual number of runs observed. Compare \(Z\) to the critical value from the standard normal distribution for \(\alpha = 0.05\).

Make a Decision

If \(\left| Z \right|\) is greater than the critical value (approximately 1.96 for \(\alpha = 0.05\)), reject the null hypothesis and conclude the sequence is not random. Otherwise, fail to reject the null hypothesis and conclude randomness.

Key concepts

Understanding Hypothesis Testing

Hypothesis testing is a statistical method that allows us to make inferences about a population using sample data. It involves two competing hypotheses:

• The null hypothesis (H_0), which represents a statement of no effect or no difference. For our context of randomness, it means the observed data is consistent with a random process.
• The alternative hypothesis (H_a), which suggests that there is an effect or a difference. In our case, it would mean the data shows a non-random pattern.

The goal of hypothesis testing is to determine which hypothesis is more likely given the data we observe. We achieve this by calculating a test statistic, which is compared to a critical value derived from a statistical distribution. If the test statistic falls within a certain range (often determined by a significance level like 1 = 0.05), we either reject or fail to reject the null hypothesis. This process involves a balance of evidence, with the critical value serving as a threshold. When learning hypothesis testing, focus on understanding the framework and using it as a systematic approach to make decisions using data.

Demystifying Probability

Probability is a fundamental concept in statistics that quantifies the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates an impossibility and 1 indicates certainty. Understanding probability is crucial in making informed conclusions from hypothesis testing because it underlies the entire process.

Let's look at the daily lottery numbers example. By determining whether numbers are odd or even, and then deciding if the sequence is random, we effectively use probability to assess the likelihood of observing such a sequence. This can be achieved by examining how often certain patterns occur compared to what would be expected under completely random circumstances.

• For instance, if a number sequence seems to have more odd numbers than what we'd expect, probability helps us quantify how unusual that is.
• It provides a quantitative measure for the runs test used in randomness testing by allowing us to calculate the probability of observing certain numbers of runs.

Always remember that probability forms the backbone of statistical inference, enabling us to make predictions and conclusions from data.

The Nature of Randomness

Randomness refers to the unpredictability and lack of pattern in sequences/events. It is at the core of many statistical analyses, particularly in hypothesis testing. Each number in a random sequence is independent and unpredictability means that there is no discernible pattern.

Randomness may seem counterintuitive as humans often seek patterns, even when they do not exist. The runs test conducted in the original exercise aims to discern whether the sequence of odds and evens is truly random or if an underlying pattern (non-randomness) exists. For an unbiased outcome of events, such as lottery results, randomness ensures fair probabilities for all possible outcomes.

• Real-world Implications: For instance, in the context of lotteries, true randomness is important to ensure no bias towards specific number combinations.
• Statistical Tests: By using tools like the runs test, statisticians can evaluate the randomness of data, helping identify any deviations from expected variability.

Understanding randomness is crucial because so much of statistical analysis is built upon the assumption of random samples. Hence, testing for randomness helps validate our assumptions in hypothesis testing.