2022 MI P2 Q10
2022 MI P2 Q10
12 marks
Prelims
A marine biologist claims that the mean mass of an adult sockeye salmon is at least \(4.2\)kg. The mass, \(x\)kg, of each adult sockeye salmon in a random sample of \(100\) sockeye salmon is weighed. The results are summarised as follows.
\(\sum{\left( x-4.2 \right)}=-7.84\) \(\sum{{{\left( x-4.2 \right)}^{2}}=\mathrm{32}\mathrm{.186}}\)
- Find unbiased estimates of the population mean and variance of the mass of adult sockeye salmons.[3]
- Test, at the \(5\%\) level of significance, whether the marine biologist’s claim is valid.[4]
- Explain, in the context of the question, the meaning of ‘at the \(5\%\) level of significance’.[1]
- Explain, without further calculation, whether the conclusion of the test in part (ii) will remain the same if the test were conducted with the same data but with only the following changes.
- The marine biologist claims that the mean mass of an adult sockeye salmon is \(4.2\) kg.[2]
- The population variance is known and is smaller than the unbiased estimate of the population variance found in part (i).[2]
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Answer:(i) \({\mathrm{0}\mathrm{.319}}\) (\(3\) s.f.) (ii) \({p}\)-value\(=\mathrm{0}\mathrm{.0825}\)(\(3\) s.f.) Since \({p}\)-value\(=\mathrm{0}\mathrm{.0825 0}\mathrm{.05}\), we do not reject \({{H}_{0}}\). There is insufficient evidence at the 5% level of significance that the population mean mass of adults sockeye salmons is less than \({4.2}\) kg (i.e. insufficient evidence that the claim is not valid). (iii) \({\mathrm{0}\mathrm{.05}}\) is the probability of concluding that the mean mass of sockeye salmon is less than \({4.2}\) kg, when in fact it is \({4.2}\) kg. (iv)(a) The new \({p}\)-value\(=\mathrm{0}\mathrm{.165040}\mathrm{.05}\). \({{H}_{0}}\) will still not be rejected and the conclusion of the test in part (ii) will remain the same, that is, the conclusion will still be that there is insufficient evidence that the claim is not valid. (ii) \({{H}_{0}}\)may \text{ or } may not be rejected. The conclusion may \text{ or } may not be that there is insufficient evidence that the claim is not valid
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