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Suppose we wanted to see if a particular antibiotics average time (in days) to completely treat an infection was shorter than another antibiotic. Let A be the first antibiotic, and B be the second.

(a) State H0, HA.

(b) Interpret a Type I Error in terms of the problem.

(c) Interpret a Type II error in terms of the problem.

(d) Would you use α = 0.10, 0.05, or 0.01 for your hypothesis test? Explain.

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User Ouadie
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Final answer:

To compare two antibiotics' average time to treat an infection, one would formulate a null hypothesis (H0) stating that the first antibiotic is not quicker than the second and an alternative hypothesis (HA) indicating the opposite. A Type I Error would incorrectly conclude antibiotic A is faster, while a Type II Error would incorrectly conclude there is no difference when A is indeed faster. The significance level (α) reflects the tolerance for Type I Error, with lower α for less tolerance.

Step-by-step explanation:

Understanding Hypothesis Testing

To compare the average time (in days) it takes for two antibiotics to treat an infection, we can formulate the following hypotheses:

  • H0 (Null Hypothesis): The average time for antibiotic A to treat an infection is greater than or equal to that of antibiotic B. (H0: μA ≥ μB).
  • HA (Alternative Hypothesis): The average time for antibiotic A to treat an infection is less than that of antibiotic B. (HA: μA < μB).

Type I Error occurs if we reject H0 when it is true. In this context, it means we would conclude that antibiotic A is faster than B when in reality, it is not.

Type II Error occurs if we fail to reject H0 when HA is true. This means we would conclude that antibiotic A is not faster than B when it actually is.

The choice of α (alpha), the level of significance, depends on the consequences of making errors. Common choices are α = 0.10, 0.05, or 0.01. A lower α reflects a lower tolerance for making a Type I Error.

In medical contexts, where the consequences of errors can be serious, a more conservative α such as 0.01 is often chosen. However, if the consequences of a Type I Error are less severe and we prefer more power to detect a difference when there is one, a higher α such as 0.05 or 0.10 may be reasonable.

User BClaydon
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