In a drinks factory, a machine is programmed to dispense $500$ ml of green tea into empty bottles.

Question

After a routine check of the machine, the production manager suspects that the machine is dispensing more green tea than expected. A random sample of $50$ bottles is taken for a hypothesis test at $\alpha \%$ level of significance and the data is as follows:
Taking $\alpha =2$, test the production manager’s suspicion and state the meaning of the $p$-value in the context of the question.[5]
Volume of green tea (correct to nearest ml)
$498$
$499$
$500$
$501$
$502$
$503$
$504$
Number of bottles
$5$
$11$
$10$
$11$
$10$
$1$
$2$
1. Determine the set of values of $\alpha $ for which the production manager’s suspicion is valid.[1]
2. Explain why the production manager is able to carry out the hypothesis test without knowing anything about the distribution of the volume of green tea dispensed by the machine.[1]
The machine is being recalibrated to dispense $500$ ml of green tea. Another random sample of $50$ bottles of green tea is taken and the mean and standard deviation of this sample are $502$ ml and $k$ ml respectively. A hypothesis test, at the $5\%$ significance level, concluded that the recalibration is done accurately. Find the set of values that $k$ can take.[5]

Accepted Answer

(a) From GC, ${p	ext{-value}}$ is ${0.0259}$. (3 s.f.) Since ${p	ext{-value} = 0.0259 > 0.02}$, we do not reject ${H_0}$ at ${2\%}$ level of significance and conclude that there is insufficient evidence that the production manager's suspicion is valid. The ${p	ext{-value}}$ is ${0.0259}$ and it means there is a probability of ${0.0259}$ of observing a test statistic, ${z \geq 1.95}$, given that the population mean green tea dispensed in a bottle is ${500	ext{ ml}}$. (i) ${2.59 7.14}$ (3 s.f.)

2024 JPJC P2 Q10

2024 JPJC P2 Q10

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