What Are The Key Terms In A One-Sample Verbal Hypothesis

What Are The Key Terms In A One Sample Verbal Hypothesis That Signify

The assignment asks: What are the key terms in a one-sample verbal hypothesis that indicate whether a one-tailed or two-tailed test is being conducted? Additionally, it requires providing one example each for a one-tailed and a two-tailed test, including the null and alternative hypotheses in symbolic form.

Furthermore, the assignment presents a contextual scenario involving Newark Liberty Airport's runway operations, jet arrival rates, and associated waiting costs. It involves calculating the total hourly waiting costs under different operational policies: normal operation with both runways, shutdown of one runway due to repair, and enhanced policy with both runways available for both jet types.

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The key terms in a one-sample verbal hypothesis that indicate whether a test is one-tailed or two-tailed are primarily centered around the language used to describe the hypothesized difference or relationship in the population parameter. In statistical hypothesis testing, the verbal hypotheses express claims about a parameter—such as a mean, proportion, or rate—and the way these claims are phrased determines whether the test is one-tailed or two-tailed.

In a one-tailed test, the verbal hypothesis explicitly states that the parameter is either greater than or less than the hypothesized value. Words such as "greater than," "less than," "at least," or "at most" are indicative of a one-tailed hypothesis. For example, a hypothesis might be expressed as, "The average waiting time for jets exceeds 30 minutes," suggesting a one-tailed test focused on whether the mean is greater than 30.

Conversely, a two-tailed test is indicated by a verbal hypothesis that the parameter is "different from" or "not equal to" a certain value, implying interest in deviations in either direction. Phrases such as " differs from," "is not equal to," or "is different from" signify a two-tailed hypothesis. For example, "The average waiting time for jets is different from 30 minutes," signals a two-tailed test where deviations in both directions are examined.

Examples:

• One-tailed test: Null hypothesis (H₀): "The mean waiting time is less than or equal to 30 minutes," or in symbols, H₀: μ ≤ 30. The alternative hypothesis (H₁): "The mean waiting time exceeds 30 minutes," i.e., H₁: μ > 30.
• Two-tailed test: Null hypothesis (H₀): "The mean waiting time is equal to 30 minutes," or in symbols, H₀: μ = 30. The alternative hypothesis (H₁): "The mean waiting time is not equal to 30 minutes," i.e., H₁: μ ≠ 30.

The contextual scenario involves Newark Liberty Airport, where two runways serve different jet sizes with specified arrival rates and costs. First, the total hourly waiting costs must be calculated considering both large and small jets, their arrival rates, and circling costs. Given the arrival rates—20 large jets per hour on runway 1R, and 30 small jets per hour on runway 1L—the total waiting costs are predicated upon whether both runways operate simultaneously or if one is shut down.

When both runways are operational, each handles specific types of jets. The wait times can be derived considering the arrival rates and maximum capacity of 60 jets per hour per runway. Because the arrival rates are well below capacity, the waiting times relate to the excess demand beyond capacity. The cost per minute for waiting varies by jet size, adding to the total cost calculations.

Scenario 1: Both runways are functional. The total waiting time for each jet type is calculated based on excess demand beyond the capacity, then multiplied by the respective costs ($25/min for large jets and $10/min for small jets). Summing these yields the total hourly waiting costs.

Scenario 2: When runway 1L is shut down temporarily and all jets land on runway 1R, the total waiting time and costs increase because the single runway must accommodate all jets. Since the combined arrival rate (20 + 30 = 50 jets per hour) remains below the capacity of 60 jets per hour, the waiting time reduces, but the total cost must be recalculated accordingly.

Scenario 3: After repair, both runways reopen and are available for any jet type. This scenario allows for optimized runway usage, potentially decreasing total waiting times and costs by balancing the load across both runways, reducing waiting durations for more expensive large jets, and minimizing overall costs.

This detailed analysis from a hypothesis testing perspective involves comparing different operational policies' outcomes, possibly testing whether the waiting times or costs significantly differ between configurations. The hypotheses could be framed as statements about the mean waiting time or total costs under different scenarios, with the type of test (one-tailed or two-tailed) chosen based on the specific research question—whether testing for an increase, decrease, or simply a difference.

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