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Worked solution

Define a random sample and use shifted totals

A production manager wants a random quality-control sample of one type of chocolate bar from the thousands produced at a factory in one day. For 70 sampled bars, mass xx grams satisfies ∑(x−95)=287\sum(x-95)=287 and ∑(x−95)2=1928\sum(x-95)^2=1928.

(i) State what random sampling means in this context. mark

(ii) Calculate unbiased population mean and variance estimates. marks

[2020/NYJC/II/11(part)]

CJC · Review problems · Question 2

Answer

(i) Equal-probability simple random samples of 70 bars from that day’s production. (ii) Mean 99.199.1 g; variance 10.910.9 g².

(i) State the sampling condition in context

Every bar of that type made during that day has an equal chance of inclusion; for a simple random sample, every possible subset of 70 such bars has the same chance of being selected. The latter statement is stronger than equal individual inclusion probabilities alone and retains the source’s full definition.

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(ii) Calculate the shifted estimates

xˉ=95+287/70=99.1\bar x=95+287/70=99.1 g. Subtracting 95 does not change variance, so s2=169(1928−287270)=10.8884058…≈10.9s^2=\dfrac1{69}\left(1928-\dfrac{287^2}{70}\right)=10.8884058\ldots\approx10.9 g².

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Common mistakes

Equal individual inclusion chances alone do not guarantee that every size 70 subset is equally likely.

Read the hint