A company has a machine designed to fill bags with, on average, ${{\mu }_{0}}$ kg of salt. The mas

Question

A company has a machine designed to fill bags with, on average, ${{\mu }_{0}}$ kg of salt. The mass of salt in a randomly chosen bag has a normal distribution with population standard deviation denoted by $\sigma $ kg. The production manager wishes to investigate if the machine is adjusted correctly. He takes a random sample of $n$ bags and carries out a hypothesis test at the $1\%$ level of significance. State null and alternative hypotheses for the manager’s test, defining any parameters you use.[2] Find, in terms of ${{\mu }_{0}}$, $\sigma $ and $n$, the critical region(s) for this test.[4] The company has a different machine which fills bags with, on average, $25$kg of low sodium salt. One of the company’s production supervisor has reported that some of the workers suspect the machine is no longer set correctly, and the average mass of low sodium salt in the bags may in fact be more than $25$kg.The production supervisor decides to carry out a hypothesis test at the $0.5\%$ level of significance with a random sample of $80$ bags of low sodium salt. Summary data for the mass, $y$ kg, of low sodium salt in these bags is as follows. ${n = 80} \qquad \sum(y - 25) = 27.2 \qquad \sum(y - 25)^2 = 85.1$ Carry out the test and state the conclusion of the test in the context of the question.[5] State, giving a reason, whether any assumptions about the population are needed in order for the test to be valid.[1]

Accepted Answer

(a)(i) ${{H}_{0}}:\mu ={{\mu }_{0}}$; ${{H}_{1}}:\mu e {{\mu }_{o}}$ (ii) $\left( 0,{{\mu }_{0}}-\frac{2.58\sigma }{\sqrt{n}} \right]\cup \left[ {{\mu }_{0}}+\frac{2.58\sigma }{\sqrt{n}},\infty \right)$ (b)(i) Using a $z$-test, $p$-value $\mathrm{=P}\left( \bar{Y}\ge 25.34 \right)\approx 0.000956$ Since $p$-value = $0.000956$ (3 s.f.) < $0.005$, we reject ${{H}_{0}}$ and conclude there is sufficient evidence, at the $0.5%$ level of significance, that the mean mass of low sodium salt in the bags is more than $25$ kg. (ii) No assumption about the population is needed. Since sample size ${80}$ is large, by Central Limit Theorem, ${\overline{Y}}$ follows a normal distribution approximately.

2024 RI P2 Q10

2024 RI P2 Q10

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