One-tailed test (increase or decrease)

A question is this sub-type if and only if it provides observed data and asks to perform a complete hypothesis test where the alternative hypothesis is directional (testing for an increase OR a decrease in the mean rate), including stating hypotheses, comparing to critical value or p-value, and reaching a conclusion.

54 questions · Standard +0.4

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AQA S3 2007 June Q7
12 marks Standard +0.8
7 In a town, the total number, \(R\), of houses sold during a week by estate agents may be modelled by a Poisson distribution with a mean of 13 . A new housing development is completed in the town. During the first week in which houses on this development are offered for sale by the developer, the estate agents sell a total of 10 houses.
  1. Using the \(10 \%\) level of significance, investigate whether the offer for sale of houses by the developer has resulted in a reduction in the mean value of \(R\).
  2. Determine, for your test in part (a), the critical region for \(R\).
  3. Assuming that the offer for sale of houses on the new housing development has reduced the mean value of \(R\) to 6.5, determine, for a test at the 10\% level of significance, the probability of a Type II error.
    (4 marks)
Edexcel S4 Q5
12 marks Standard +0.3
5. (a) Define
  1. a Type I error,
  2. a Type II error. A small aviary, that leaves the eggs with the parent birds, rears chicks at an average rate of 5 per year. In order to increase the number of chicks reared per year it is decided to remove the eggs from the aviary as soon as they are laid and put them in an incubator. At the end of the first year of using an incubator 7 chicks had been successfully reared.
    (b) Assuming that the number of chicks reared per year follows a Poisson distribution test, at the \(5 \%\) significance level, whether or not there is evidence of an increase in the number of chicks reared per year. State your hypotheses clearly.
    (c) Calculate the probability of the Type I error for this test.
    (d) Given that the true average number of chicks reared per year when the eggs are hatched in an incubator is 8, calculate the probability of a Type II error.
AQA Further AS Paper 2 Statistics 2018 June Q7
8 marks Standard +0.3
7 Over a period of time it has been shown that the mean number of vehicles passing a service station on a motorway is 50 per minute. After a new motorway junction was built nearby, Xander observed that 30 vehicles passed the service station in one minute. 7
  1. Xander claims that the construction of the new motorway junction has reduced the mean number of vehicles passing the service station per minute. Investigate Xander's claim, using a suitable test at the \(1 \%\) level of significance.
    7
  2. For your test carried out in part (a) state, in context, the meaning of a Type 1 error. 7
  3. Explain why the model used in part (a) might be invalid.
OCR MEI Further Statistics Major Specimen Q8
12 marks Standard +0.3
8 Natural background radiation consists of various particles, including neutrons. A detector is used to count the number of neutrons per second at a particular location.
  1. State the conditions required for a Poisson distribution to be a suitable model for the number of neutrons detected per second. The number of neutrons detected per second due to background radiation only is modelled by a Poisson distribution with mean 1.1.
  2. Find the probability that the detector detects
    (A) no neutrons in a randomly chosen second,
    (B) at least 60 neutrons in a randomly chosen period of 1 minute. A neutron source is switched on. It emits neutrons which should all be contained in a protective casing. The detector is used to check whether any neutrons have not been contained; these are known as stray neutrons. If the detector detects more than 8 neutrons in a period of 1 second, an alarm will be triggered in case this high reading is due to stray neutrons.
  3. Suppose that there are no stray neutrons and so the neutrons detected are all due to the background radiation. Find the expected number of times the alarm is triggered in 1000 randomly chosen periods of 1 second.
  4. Suppose instead that stray neutrons are being produced at a rate of 3.4 per second in addition to the natural background radiation. Find the probability that at least one alarm will be triggered in 10 randomly chosen periods of 1 second. You should assume that all stray neutrons produced are detected.