91Ó°ÊÓ

Presidents: Party Affiliation For each successive presidential term from Teddy Roosevelt to George W. Bush (first term), the party affiliation controlling the White House is shown below, where R designates Republican and D designates Democrat. (Reference: The New York Times Almanac.) \(\begin{array}{llllllllllllllllllll}\mathrm{R} & \mathrm{R} & \mathrm{R} & \mathrm{D} & \mathrm{D} & \mathrm{R} & \mathrm{R} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{D} & \mathrm{R} & \mathrm{D} & \mathrm{R} & \mathrm{R} & \mathrm{D} & \mathrm{R} & \mathrm{R} & \mathrm{R} & \mathrm{D} & \mathrm{D} & \mathrm{R}\end{array}\) Historical Note: In cases in which a president died in office or resigned, the period during which the vice president finished the term is not counted as a new term. Test the sequence for randomness. Use \(\alpha=0.05\).

Step by step solution

01

Define Expected Outcomes

In a random sequence, the transitions between parties should occur infrequently to demonstrate randomness. We are testing whether the observed number of such transitions significantly deviates from what is expected under randomness. Each party transition (R to D or D to R) is counted as one run.

02

Count Party Transitions

Analyze the sequence and count the total number of alternations, or 'runs', from one party to another. For the sequence given, we identify the transitions between parties: R to D or D to R. Upon examining the sequence of party affiliations, we find 10 transitions (runs): RR, R, DDD, R, DD, RRR, D, and R. Thus, there are 10 runs.

03

Calculate Expected Number of Runs

Calculate the expected number of runs (E) and variance (Var) for such a sequence of binary outcomes. The expected number of runs in a two-outcome sequence is given by: \[ E(R) = \frac{2n_1n_2}{n} + 1 \]Where \( n_1 \) is the number of R's, \( n_2 \) is the number of D's, and \( n \) is the total number of terms. In this case, \( n_1 = 11 \), \( n_2 = 12 \), and \( n = 23 \). Thus, \[ E(R) = \frac{2 \times 11 \times 12}{23} + 1 = 12.48 \]

04

Variance of the Run Count

The variance of the number of runs is calculated as:\[ \text{Var}(R) = \frac{2n_1n_2(2n_1n_2-n)}{n^2(n-1)} \]Substitute values: \[ \text{Var}(R) = \frac{2 \times 11 \times 12(2 \times 11 \times 12 - 23)}{23^2 \times (23-1)} = 5.89 \]

05

Calculate Test Statistic

The test statistic Z is calculated as:\[ Z = \frac{R - E(R)}{\sqrt{\text{Var}(R)}} \]Given \( R = 10 \), \( E(R) = 12.48 \), and \( \text{Var}(R) = 5.89 \), substitute these values:\[ Z = \frac{10 - 12.48}{\sqrt{5.89}} = -1.03 \]

06

Decision Rule and Conclusion

Compare the calculated Z value with the critical Z value from the standard normal distribution at \( \alpha = 0.05 \), which is approximately \( \pm 1.96 \). Since \(|Z| = 1.03\) is less than 1.96, we cannot reject the null hypothesis, indicating the sequence does not deviate significantly from randomness.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

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

Randomness Test

A randomness test is performed to determine if a sequence behaves in a way that suggests random selection, rather than following a detectable pattern. It is important because this test helps assess the unpredictability of sequences, especially when trying to infer if the sequence occurs just by chance.

• In the context of political party shifts in the White House, testing for randomness helps to check whether changes in party control happen by chance or follow a more systematic, predictable pattern.
• This can assist in exploring whether historical patterns of political shifts have genuine stochastic behavior or are affected by underlying factors such as socio-political trends.

Other situational applications could involve similar tests in fields such as genetics, quality control in manufacturing, or finance, wherever random patterns are expected or assumed.

Party Affiliation Sequence

The party affiliation sequence refers to the historical listing of political parties in control, represented here by 'D' for Democrat and 'R' for Republican. This is a binary sequence because it consists of only two possible outcomes—either R or D.

• The sequence can be significant as it portrays how often political control has shifted over time.
• Some might analyze this pattern to predict future political landscapes or to understand the historical frequency of political shifts.

Understanding this sequence means observing the series of transitions from Democrat to Republican and vice versa within specified intervals, in this case, presidential terms.
In essence, it presents a concise history of political party dominance that can aid in evaluating political trends.

Statistical Significance

Statistical significance in the context of hypothesis testing refers to the probability of the observed result occurring under the null hypothesis. It helps determine whether the outcome can be attributed to chance or indicates an actual effect or pattern.

• For this sequence of party affiliations, the statistical significance is assessed through the calculated test statistic and comparison to a critical value.
• This step determines whether the observed number of transitions (observed in the sequence) significantly deviates from what would be expected by pure chance."

Bearing statistical significance implies that results are unlikely to have arisen randomly and possibly reflect real-world trends or effects, making it a core component of interpreting data in this randomness test.

Runs Test

The Runs Test is a non-parametric test used to determine whether a sequence is random. It focuses on identifying 'runs'—consecutive identical items (e.g., RRR or DD), and examines their number relative to what is statistically expected in a random sequence.

• More formally, this involves comparing the observed number of runs with the expected number, considering the sequence's length and composition (number of R's and D's).
• A run represents a crucial structural element in this context; too few or too many runs compared to the expected can signal non-randomness.

Runs Test calculates a Z-statistic to discern if the sequence deviates from randomness. The computed Z falls within a standard normal distribution reference range to make inferences, thereby supporting or refuting the null hypothesis of randomness. In this case, the calculated Z=-1.03 is within the acceptable range indicating randomness.