For quality control, the production manager of a company wishes to take a random sample of a certain

Question

For quality control, the production manager of a company wishes to take a random sample of a certain type of chocolate bar from his factory. He wants to check that the mean mass of the bars is $52$ grams, as stated on the packets.

State what it means for a sample to be random in this context.[1]

The masses, $x$ grams, of a random sample of $80$ chocolate bars are summarised as follows.

${n = 80} \qquad \sum(x - 52) = -37 \qquad \sum(x - 52)^2 = 310.7$

Calculate the unbiased estimates of the population mean and variance.[2]
What do you understand by the term ‘unbiased estimate’?[1]
Carry out a suitable hypothesis test, explaining the choice between a $1$-tail test and a $2$-tail test.
You should state your hypotheses and define any symbols you use.
Referring to the $p$-value for your test, explain what it indicates.[6]
Explain whether there is a need for the manager to know anything about the population distribution of the masses of the chocolate bars.[2]

Accepted Answer

Answers: (a) Each chocolate bar in the population has an equal chance of being chosen and the chocolate bars are chosen independently. (b) ${3.72}$ (3 s.f.) (c) An estimate is unbiased when the expected value of the estimator used to obtain the estimate is equal to the value of the population parameter. (d) From GC, ${p-\mathrm{velue} = 0.0319}$ (3 s.f.) It indicates that if the level of significance is ${3.19\%}$ or more, the null hypothesis (population mean mass is ${52	ext{ grams}}$) will be rejected. Otherwise, the null hypothesis will not be rejected. (e) There is **no need** for the manager to know anything about the population distribution of the masses of the chocolate bars since ${\bar{X} \sim \mathrm{N}\left(\mu, \frac{s^2}{n}ight)}$ **approximately** by Central Limit Theorem as ${n = 80}$ is large and ${z	ext{-test}}$ can be used.

2024 DHS P2 Q10

2024 DHS P2 Q10

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