Confidence intervals from small samplesAQA A-Level Further Maths: Subtopic test

1. 1
   The masses of chocolate bars from a production line are normally distributed. A random sample of $8$ bars has sample mean $36.2$ g and an unbiased estimate of the population standard deviation of $2.4$ g.

   (a)

   State the number of degrees of freedom of the $t$-distribution used to construct a confidence interval for the population mean.

   [1 mark]

   • A$8$
   • B$6$
   • C$9$
   • D$7$

   (b)

   The $2.5\%$ critical value of $t$ with $7$ degrees of freedom is $2.365$. Find a $95\%$ confidence interval for the population mean mass.

   [1 mark]

   • A$(34.54,\ 37.86)$
   • B$(34.19,\ 38.21)$
   • C$(34.24,\ 38.16)$
   • D$(30.52,\ 41.88)$

   (c)

   The $0.5\%$ critical value of $t$ with $7$ degrees of freedom is $3.499$. Find a $99\%$ confidence interval for the population mean mass.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The battery life, in hours, of a make of phone is normally distributed. A random sample of $12$ phones has $\sum x=220.8$ and $\sum x^2=4087.47$.

   (a)

   Find an unbiased estimate of the population variance.

   [1 mark]

   • A$2.25$ hours$^2$
   • B$2.06$ hours$^2$
   • C$1.50$ hours$^2$
   • D$0.19$ hours$^2$

   (b)

   The $5\%$ critical value of $t$ (one tail) with $11$ degrees of freedom is $1.796$. Find a $90\%$ confidence interval for the population mean battery life.

   [1 mark]

   • A$(17.69,\ 19.11)$
   • B$(17.45,\ 19.35)$
   • C$(17.62,\ 19.18)$
   • D$(15.71,\ 21.09)$

   (c)

   The manufacturer claims that the mean battery life is $20$ hours. Use your interval from (b) to comment on this claim.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The reaction times, in milliseconds, of a random sample of $7$ drivers are $248,\ 262,\ 255,\ 241,\ 270,\ 259,\ 253$. Reaction times may be assumed to be normally distributed.

   (a)

   Calculate the sample mean and an unbiased estimate of the population variance.

   [3 marks]

   (b)

   The $2.5\%$ critical value of $t$ with $6$ degrees of freedom is $2.447$. Construct a $95\%$ confidence interval for the mean reaction time.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A chemist measures the percentage purity of aspirin in a random sample of $5$ tablets from one batch: $98.1,\ 97.4,\ 98.6,\ 97.9,\ 98.0$. Purities are normally distributed.

   (a)

   (i) The $2.5\%$ critical value of $t$ with $4$ degrees of freedom is $2.776$. Construct a $95\%$ confidence interval for the mean purity.
   (ii) The manufacturer claims that the mean purity is $98.5\%$. Comment on this claim using your interval.
   (iii) Explain why a $t$-distribution is used rather than a normal distribution.

   [6 marks]

   (b)

   The chemist then takes a random sample of $25$ tablets from a second batch. The sample mean is $97.98\%$ and the unbiased estimate of the population standard deviation is $0.52\%$.
   (i) The $2.5\%$ critical value of $t$ with $24$ degrees of freedom is $2.064$. Construct a $95\%$ confidence interval for the mean purity of the second batch.
   (ii) Compare the width of this interval with that in (a)(i), and evaluate what the comparison shows.

   [6 marks]

   Total for question 4: 12 marks

End of questions