Free response · Unit 2, Unit 3 Beta

On-time flights: a decision rule, its errors and their consequences

Q4: Across several units · calculator allowed · 10 points

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Name: ______________________   Date: ____________   On-time flights: a decision rule, its errors and their consequences (10 points)

Table

Checking an airline's on-time claim

An airline advertises that 90% of its flights arrive on time. A consumer group will file a formal complaint if it finds convincing evidence that fewer than 90% of the airline's flights arrive on time. The group selects a random sample of 150 of the airline's thousands of flights from last month and records whether each arrived on time. The group's rule: file the complaint if 128 or fewer of the 150 flights arrived on time.

The group's plan
ItemValue
Advertised on-time proportion0.90
Flights sampled150, chosen at random
Decision rulecomplain if 128 or fewer arrived on time

  1. (a) Suppose the airline's claim is true. Let XX be the number of the 150 sampled flights that arrive on time. Identify the distribution of XX, with its parameters, and calculate P(X≤128)P(X \le 128). [2 points]

  2. (b) The group's rule is a significance test. Identify the null and alternative hypotheses, defining the parameter. [1 point]

  3. (c) Describe a Type I error in the context of this study, and explain one consequence of making it. [2 points]

  4. (d) Describe a Type II error in the context of this study, and explain one consequence of making it. [2 points]

  5. (e) Interpret the probability you calculated in part (a) in terms of the group's possible errors. [1 point]

  6. (f) The group wants a smaller probability of a Type II error without raising the probability of a Type I error. Explain how it could change its study to do this, and why the change works. [2 points]