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.

3 questions · Standard +0.5

5.02i Poisson distribution: random events model
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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 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 \(\_\_\_\_\)