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

1. 1
   The mass of a packet of flour, in grams, is Normally distributed with standard deviation $6$. A random sample of $25$ packets has mean mass $503.2$ g.

   (a)

   Find the standard error of the sample mean.

   [1 mark]

   • A$0.24$
   • B$1.2$
   • C$6$
   • D$1.44$

   (b)

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

   [1 mark]

   • A$(502.0,\ 504.4)$
   • B$(491.4,\ 515.0)$
   • C$(500.85,\ 505.55)$
   • D$(501.23,\ 505.17)$

   (c)

   The label says the mean mass is $500$ g. Use your interval from part (b) to comment on this claim, explaining the link with a hypothesis test.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A random sample of $36$ observations is taken from a Normal distribution with standard deviation $2.4$. The sample mean is $13.0$.

   (a)

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

   [1 mark]

   • A$(12.34,\ 13.66)$
   • B$(12.22,\ 13.78)$
   • C$(9.05,\ 16.95)$
   • D$(12.49,\ 13.51)$

   (b)

   Which statement is the correct interpretation of the $90\%$ confidence interval?

   [1 mark]

   • AThere is a $90\%$ probability that $\mu$ lies between $12.34$ and $13.66$
   • B$90\%$ of the observations lie between $12.34$ and $13.66$
   • CThe sample mean lies in the interval with probability $0.9$
   • DIf many samples of $36$ were taken and an interval found from each in the same way, about $90\%$ of those intervals would contain $\mu$

   (c)

   The sample size is to be increased so that the width of the $90\%$ confidence interval is at most $0.8$. Find the smallest sample size required.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time of one machine cycle, in seconds, is Normally distributed with standard deviation $0.5$. The mean of a random sample of $40$ cycles is $12.84$ s.

   (a)

   Calculate a $99\%$ confidence interval for the mean cycle time.

   [3 marks]

   (b)

   The manufacturer claims the mean cycle time is $13.0$ s, and a rival claims it is $12.6$ s. Use your interval to decide whether a two-tailed test at the $1\%$ level would reject each claim. Explain the link between the interval and the test, and confirm your conclusion for the rival's claim with a test statistic.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lifetime of a type of battery, in hours, is Normally distributed with standard deviation $18$. A random sample of $30$ batteries has mean lifetime $412$ hours.

   (a)

   (i) Calculate a $95\%$ confidence interval for the mean lifetime.
   (ii) The manufacturer claims that the mean lifetime is $420$ hours. Use your interval to comment on this claim.

   [6 marks]

   (b)

   The $95\%$ interval from a sample of $30$ is judged too wide. Find the smallest sample size for which a $95\%$ interval has width at most $10$ hours. Evaluate the effect on the width, for $n=30$, of using a $99\%$ interval instead of a $95\%$ interval, and comment on the trade-off.

   [6 marks]

   Total for question 4: 12 marks

End of questions