Populations, censuses and samplesEdexcel International A Level Further Maths: Subtopic test

1. 1
   A council keeps a register of the $4200$ households in its area. To find out how many households recycle, it selects $300$ households from the register and sends each one a questionnaire.

   (a)

   The register of households is an example of

   [1 mark]

   • Aa population
   • Ba sampling frame
   • Ca census
   • Da statistic

   (b)

   In this survey, the sampling unit is

   [1 mark]

   • Aone household
   • Bthe $4200$ households
   • Cthe $300$ households
   • Dthe council

   (c)

   Give one advantage and one disadvantage of the council using a sample of $300$ households rather than a census.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The masses of bags of flour are modelled by $\mathrm{N}(\mu,\sigma^2)$, where $\mu$ and $\sigma$ are unknown. A random sample $X_1,X_2,X_3,X_4,X_5$ of bags is taken and $\bar X$ is the sample mean.

   (a)

   Which of the following is a statistic?

   [1 mark]

   • A$\bar X-\mu$
   • B$\dfrac{\bar X}{\sigma}$
   • C$\max\{X_1,X_2,X_3,X_4,X_5\}$
   • D$\mu+X_1$

   (b)

   The sampling distribution of $\bar X$ is

   [1 mark]

   • Athe distribution of the masses of all bags of flour
   • Bthe distribution of the five values in one particular sample
   • Cthe distribution of the value of $\mu$
   • Dthe probability distribution of the values of $\bar X$ over all possible samples of size $5$

   (c)

   Explain why $\bar X$ has a sampling distribution but $\mu$ does not.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bag contains three discs numbered $1$, $2$ and $3$. A disc is drawn at random and replaced, and then a second disc is drawn at random. Let $M$ be the larger of the two numbers drawn (or the common value if they are equal) and let $\bar X$ be the mean of the two numbers.

   (a)

   Find the sampling distribution of $M$.

   [3 marks]

   (b)

   Find the sampling distribution of $\bar X$, and hence find $\mathrm{P}(\bar X\ge2)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory produces $5000$ fuses each day, each marked with a serial number from $1$ to $5000$. A fuse is tested by passing a rising current through it until it blows, and it fails the quality standard if it blows below a set current. Each day the manager tests a random sample of $40$ fuses and records $\hat p$, the proportion of the sample that fail the quality standard.

   (a)

   (i) Explain why the manager does not test every fuse.
   (ii) State the sampling frame, and describe how a random sample of $40$ fuses could be chosen from it.
   (iii) Explain why $\hat p$ is a statistic, and why it has a sampling distribution.

   [6 marks]

   (b)

   On one day, $5\%$ of all the fuses produced fail the quality standard. Assume that the number $X$ of fuses in the sample of $40$ that fail is distributed $\mathrm{B}(40,0.05)$.
   (i) Find $\mathrm{P}(\hat p=0)$.
   (ii) Find $\mathrm{P}(\hat p>0.05)$.
   (iii) Use your answers to comment on how $\hat p$ relates to the proportion of defective fuses in the whole day's production.

   [6 marks]

   Total for question 4: 12 marks

End of questions