2024 ASRJC P2 Q11

2024 ASRJC P2 Q11

13 marks
  1. The leaves of a particular plant species have an average length of \(12\text{ cm}\) with a standard deviation of \(3.5\text{ cm}\). If a random sample of \(100\) leaves is selected, estimate the probability that their total length is at least \(1138\text{ cm}\).[2]
  2. An operator of a public workspace at location A claims that users of its one-seater pods spend an average of \(131\) minutes using the facilities. To test this claim, a random sample of \(64\) users was observed, revealing a mean usage time of \(127\) minutes with a standard deviation of \(16.4\) minutes.
    1. Test at \(3\%\) level of significance whether the workspace operator's claim is overstated. You should state the hypotheses and define any symbols you use.[5]
    2. Explain the meaning of '\(3\%\) level of significance' in the context of the question.[1]

    3. The workspace operator at location B claims that the mean time spent by users of its one-seater pods is \(140\) minutes, with a known population standard deviation of \(20.1\) minutes. A new sample of \(15\) pod users is taken, and the sample mean usage time, \(\bar{w}\), is reported. A hypothesis test is conducted at a \(5\%\) significance level, and the operator's claim is not rejected.

      State two necessary assumptions for the test and determine the range of values that \(\bar{w}\) can take. Give your answer correct to one decimal place.

      [5]
Similar questions are unavailable for this question.
Answer:Answers: (a) \({0.962}\) (3 s.f.) (b)(i) Let \({Y}\) denote the time spent in minutes using the one-seater pod facilities by a randomly chosen user at location \({A}\) and \({\mu}\) denote the population mean time spent in minutes using the one-seater pod facilities at location \({A}\). To test \({H_0 : \mu = 131}\) Against \({H_1 : \mu < 131}\) (Workspace operator overstating the claim) Using GC, \({p\text{-value} = \approx 0.0264}\) Since \({p\text{-value} = 0.0264 < 0.03}\), we reject \({H_0}\). There is sufficient evidence at \({3\%}\) level of significance to conclude that the centre manager was overstating his claim. (ii) There is a probability of \({0.03}\) of concluding that the population average time spent using the one-seater pod facilities at location \({A}\) is less than \({131\text{ minutes}}\) when in fact the population average time spent using the one-seater pod facilities in location \({A}\) is \({131\text{ minutes}}\). (iii) Assume that the time spent by the users of the one-seater pods facilities in location \({B}\) follows a Normal Distribution. Assume also that the time spent, on the one-seater pod facilities in location \({B}\) by users, are independent of each other. \({129.8 < \bar{w} < 150.2}\)

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