Question

Random Numbers Random? A calculator generated these integers randomly. Apply the runs test to see if you can reject the hypothesis that the numbers are truly random. Use \(\alpha=0.05 .\) $$ \begin{array}{lllllllllll} 1 & 1 & 1 & 1 & 1 & 1 & 2 & 1 & 1 & 1 & 1 \\ 2 & 2 & 1 & 2 & 1 & 2 & 2 & 1 & 2 & 1 & 1 \\ 2 & 1 & 1 & & & & & & & & \end{array} $$

Short answer

The sequence appears random; we do not reject the null hypothesis of randomness at \( \alpha = 0.05 \).

Step-by-step solution

Understand the Problem

We need to determine if the sequence of numbers given is random by using the runs test. The sequence is composed of integers 1 and 2, and our goal is to test the hypothesis on randomness at a significance level of \( \alpha = 0.05 \).

Identify the Runs

A 'run' is a sequence of identical numbers followed by a different number. The sequence provided can be parsed into runs: - Run 1: five 1s (1, 1, 1, 1, 1) - Run 2: one 2 - Run 3: one 1 - Run 4: one 2 - Run 5: one 1 - Run 6: one 2 - Run 7: two 2s (2, 2) - Run 8: two 1s (1, 1) - Run 9: one 2 - Run 10: one 1 - Run 11: two 1s (1, 1) - Run 12: one 2 - Run 13: two 1s (1, 1)

Number of Runs

Count the total number of runs in the sequence, which based on our identification is 13.

Calculate Expected Runs

Use the formula for expected number of runs: \[ E(R) = \frac{2n_1n_2}{n} + 1 \] where \( n_1 \) is the number of 1s (15), \( n_2 \) is the number of 2s (9), and \( n = 24 \) (the total number of values).\[ E(R) = \frac{2 \times 15 \times 9}{24} + 1 = 12.25 \]

Calculate Standard Deviation of Runs

The standard deviation of the number of runs \( \sigma \) is given by:\[ \sigma = \sqrt{ \frac{2n_1n_2(2n_1n_2-n_1-n_2)}{n^2(n-1)}} \]Substituting in the values provides:\[ \sigma = \sqrt{ \frac{2 \times 15 \times 9 (2 \times 15 \times 9 - 15 - 9)}{24^2 \times 23}} \approx 2.28 \]

Calculate the Z-value

Calculate the Z-value to determine the number of standard deviations the observed number of runs is from the expected number of runs:\[ Z = \frac{R - E(R)}{\sigma} \]Substituting in our values:\[ Z = \frac{13 - 12.25}{2.28} \approx 0.33 \]

Determine Critical Z-value and Decision

For a two-tailed test at \( \alpha = 0.05 \), the critical Z-value is approximately \( \pm 1.96 \). Because \( 0.33 \) is within the range of \(-1.96\) to \(1.96\), we fail to reject the null hypothesis.

Key concepts

Random Hypothesis Testing

Hypothesis testing is a fundamental concept in statistics used to determine if there is enough evidence to reject a null hypothesis. In this context, a null hypothesis is the assumption that a sequence of numbers is random. Random Hypothesis Testing aims to decide whether the observed data follows a random pattern or if there is an identifiable non-random pattern present. The Runs Test, a specific type of non-parametric test, is particularly useful for analyzing sequences consisting of two types of data points, such as the integers 1 and 2 in this exercise.

When applying the Runs Test, we analyze the "runs" within the sequence—these are uninterrupted sequences of identical numbers. A hypothesis test is conducted based on the number of runs in the data, allowing us to evaluate the randomness. If the number of observed runs significantly deviates from what we expect under randomness, we may conclude that the sequence does not behave randomly.

Significance Level Alpha

The significance level, often denoted as \( \alpha \), is a threshold used in hypothesis testing to determine whether to reject the null hypothesis. In this exercise, the significance level is set at \( \alpha = 0.05 \). This means there is a 5% risk of rejecting the null hypothesis when it is actually true.

The choice of \( \alpha \) affects how conservative or liberal the test is. A smaller \( \alpha \) (e.g., 0.01) requires stronger evidence against the null hypothesis before rejecting it, whereas a larger \( \alpha \) (e.g., 0.10) allows for a more lenient rejection threshold. In the Runs Test, the calculated Z-value is compared against critical values, and if the Z-value falls within the range determined by the significance level (in this case, between approximately -1.96 and 1.96 for a two-tailed test), we fail to reject the null hypothesis.

Expected Number of Runs

The expected number of runs, denoted as \( E(R) \), represents the number of runs we would anticipate in a sequence if it were truly random. For a sequence consisting of two types of data points, it is calculated using the formula:

\[ E(R) = \frac{2n_1n_2}{n} + 1 \]

where:\

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• \( n_1 \) is the count of the first type of data point (e.g., the number of 1s),
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• \( n_2 \) is the count of the second type of data point (e.g., the number of 2s),
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• \( n \) is the total number of observations.
\

In this exercise, we're analyzing a sequence with \( n_1 = 15 \) for 1s and \( n_2 = 9 \) for 2s, leading to an expected number of runs \( E(R) = 12.25 \). This value is crucial as it establishes the basis for comparison with the actual number of observed runs.

Standard Deviation of Runs

The standard deviation of runs, represented as \( \sigma \), helps measure the variability of the number of runs from their expected number if the sequence is random. It provides insight into the spread of the data around the expected mean. The standard deviation in the context of runs is calculated using the formula:

\[ \sigma = \sqrt{ \frac{2n_1n_2(2n_1n_2-n_1-n_2)}{n^2(n-1)}} \]

where:\

\
• \( n_1 \) and \( n_2 \) are the counts of each type of data point,
\
• \( n \) is the total number of data points.
\

For the sequence analyzed in the exercise, \( \sigma \) is approximately 2.28. This statistic is key for establishing the critical region in hypothesis testing—essentially allowing us to gauge how much deviation from the expected number of runs is significant.