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2021 ASRJC P2 Q11

2021 ASRJC P2 Q11

12 marks
Prelims

An airline which charter various flights within the United States of America has a flight route that travels from New Orleans to Miami. The airline claims that the average flight time is \(115\) minutes. Brandon, a regular commuter of this flight route, thinks that his journey takes a shorter time on average.

  1. Explain, whether Brandon should carry out a \(1\)-tail test or a \(2\)-tail test. State the hypotheses for the test, defining any symbols you use.[2]

Brandon records the flight time (in minutes) of \(8\) randomly selected flights as shown below.

\(113.0\)      \(112.4\)     \(110.0\)     \(113.8\)     \(111.3\)     \(115.2\)     \(114.2\)     \(115.5\)

  1. Assuming that the flight time from New Orleans to Miami follows a normal distribution that has a variance of \(11.3\) minutes\(^{2}\), carry out the test for Brandon at \(10\%\) level of significance. Give your conclusion in the context of the question.[3]
  2. Explain what you understand by the phrase “at \(10\%\) level of significance” in the context of the question.[1]
  3. The same sample of \(8\) flights is now used to carry out a test at the \(10\%\) significance level, to test whether the mean flight times is different from what the airline claims. Using the answer in (ii), state the conclusion of this test.[2]

Another flight route travels from Portland to Los Angeles. Due to bad weather conditions, the flights sometimes take a longer time to arrive at Los Angeles. The airline claims that the mean flight time is \(k\) minutes. The operation manager of the airline believes that this number is understated and wants to test this claim. Sixty flights were randomly selected, and the mean flight time of \(181\) minutes and standard deviation of \(13.6\) minutes were recorded.

  1. Find the range of values of \(k\) for which the claim by the airline is supported at \(4\%\) level of significance. Give your answer to the nearest minute.[4]

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Answer:(i) Let \({X}\) be the random variable denoting the flight time from New Orleans to Miami in minutes and \({\mu}\) be the mean flight time. \({H_0 : \mu = 115 \text{}\hspace{0.5em} H_1 : \mu < 115}\) (ii) Since the \({p\text{-value} < 0.1}\) (the significance level), we reject \({H_0}\) and conclude that there is sufficient evidence at the \({10\%}\) level of significance to conclude that the mean flight time from New Orleans to Miami is less than \({115\text{ minutes}}\). (iii) \({10\%}\) evel of significance refers to a probability of \({0.1}\) that we conclude that the average flight time from New Orleans to Miami is less than \({115\text{ minutes}}\) when in fact it is \({115\text{ minutes}}\). (iv) Since the \({p\text{-value} > 0.1}\) (the significance level), we do not reject \({H_0}\) and conclude that there is insufficient evidence at the \({10\%}\) level of significance to conclude that the mean flight time from New Orleans to Miami is not \({115\text{ minutes}}\).

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