Confidence intervals for a meanAQA A-Level Further Maths: Subtopic test

1. 1
   The masses of packets of porridge oats are normally distributed with standard deviation $12$ g. A random sample of $36$ packets has mean mass $504$ g.

   (a)

   Find the standard error of the sample mean.

   [1 mark]

   • A$12$ g
   • B$2$ g
   • C$0.33$ g
   • D$72$ g

   (b)

   Find a $95\%$ confidence interval for the population mean mass.

   [1 mark]

   • A$(480.48,\ 527.52)$
   • B$(502.00,\ 506.00)$
   • C$(500.08,\ 507.92)$
   • D$(500.71,\ 507.29)$

   (c)

   Find a $99\%$ confidence interval for the population mean mass.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time taken by a train to complete a journey is normally distributed with variance $9$ minutes$^2$. A random sample of $16$ journeys has mean time $42.5$ minutes.

   (a)

   Find the standard error of the sample mean.

   [1 mark]

   • A$0.5625$ minutes
   • B$3$ minutes
   • C$2.25$ minutes
   • D$0.75$ minutes

   (b)

   Find a $90\%$ confidence interval for the population mean journey time.

   [1 mark]

   • A$(41.27,\ 43.73)$
   • B$(41.03,\ 43.97)$
   • C$(37.57,\ 47.43)$
   • D$(41.75,\ 43.25)$

   (c)

   The rail company claims that the mean journey time is $45$ minutes. Use your interval from (b) to comment on this claim.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lengths of trout on a fish farm are normally distributed with standard deviation $2.4$ cm. A random sample of $64$ trout has mean length $31.2$ cm.

   (a)

   Construct a $95\%$ confidence interval for the mean length of trout on the farm.

   [3 marks]

   (b)

   The farm manager wants a $95\%$ confidence interval of width at most $0.5$ cm. Find the smallest sample size that achieves this.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A manufacturer says that the lifetime of its light bulbs is $1200$ hours on average. A tester takes a random sample of $50$ bulbs and finds a sample mean of $1172$ hours and an unbiased estimate of the population standard deviation of $96$ hours. Lifetimes may be assumed to be normally distributed.

   (a)

   (i) Construct a $95\%$ confidence interval for the mean lifetime.
   (ii) Explain why the sample standard deviation may be used in your calculation.
   (iii) Comment on the manufacturer's claim.

   [6 marks]

   (b)

   The tester takes a second random sample of $200$ bulbs, giving a sample mean of $1185$ hours and an unbiased estimate of the population standard deviation of $100$ hours.
   (i) Construct a $99\%$ confidence interval for the mean lifetime.
   (ii) Comment on whether this interval supports the manufacturer's claim, and on how it relates to the interval in (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions