Two-tailed test (change)

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 non-directional (testing whether the mean rate has changed in either direction), including stating hypotheses, comparing to critical value or p-value, and reaching a conclusion.

5 questions · Standard +0.4

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CAIE S2 2016 June Q4
6 marks Standard +0.3
4 At a certain company, computer faults occur randomly and at a constant mean rate. In the past this mean rate has been 2.1 per week. Following an update, the management wish to determine whether the mean rate has changed. During 20 randomly chosen weeks it is found that 54 computer faults occur. Use a suitable approximation to test at the \(5 \%\) significance level whether the mean rate has changed.
Edexcel S2 2016 January Q7
12 marks Standard +0.3
  1. A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution.
The fisherman takes 5 fishing trips each lasting 1 hour.
  1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,
  2. carry out the test at the \(5 \%\) level of significance. State your hypotheses clearly.
    (6) by the fisherman in an hour follows a Poisson distribution.
    The fisherman takes 5 fishing trips each lasting 1 hour.
  3. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips.
    7.
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Edexcel S2 2006 June Q4
11 marks Standard +0.3
  1. Breakdowns occur on a particular machine at random at a mean rate of 1.25 per week.
    1. Find the probability that fewer than 3 breakdowns occurred in a randomly chosen week.
    Over a 4 week period the machine was monitored. During this time there were 11 breakdowns.
  2. Test, at the \(5 \%\) level of significance, whether or not there is evidence that the rate of breakdowns has changed over this period. State your hypotheses clearly.
Edexcel S2 2013 June Q6
14 marks Standard +0.8
6. Frugal bakery claims that their packs of 10 muffins contain on average 80 raisins per pack. A Poisson distribution is used to describe the number of raisins per muffin. A muffin is selected at random to test whether or not the mean number of raisins per muffin has changed.
  1. Find the critical region for a two-tailed test using a \(10 \%\) level of significance. The probability of rejection in each tail should be less than 0.05
  2. Find the actual significance level of this test. The bakery has a special promotion claiming that their muffins now contain even more raisins. A random sample of 10 muffins is selected and is found to contain a total of 95 raisins.
  3. Use a suitable approximation to test the bakery's claim. You should state your hypotheses clearly and use a \(5 \%\) level of significance.
AQA Further Paper 3 Statistics 2023 June Q7
11 marks Standard +0.3
7 Company \(A\) uses a machine to produce toys. The number of toys in a week that do not pass Company \(A\) 's quality checks is modelled by a Poisson distribution \(X\) with standard deviation 5 The machine producing the toys breaks down.
After it is repaired, 16 toys in the next week do not pass the quality checks.
7
  1. Investigate whether the average number of toys that do not pass the quality checks in a week has changed, using the \(5 \%\) level of significance.
    7
  2. For the test carried out in part (a), state in context the meaning of a Type II error. 7
  3. Company \(B\) uses a different machine to produce toys.
    The number of toys in a week that do not pass Company B's quality checks is modelled by a Poisson distribution \(Y\) with mean 18 The variables \(X\) and \(Y\) are independent.
    Find the distribution of the total number of toys in a week produced by companies \(A\) and \(B\) that do not pass their quality checks. 7
  4. State two reasons why a Poisson distribution may not be a valid model for the number of toys that do not pass the quality checks in a week. Reason 1 \(\_\_\_\_\) Reason 2 \(\_\_\_\_\)