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

1. 1
   The masses of bags of flour are Normally distributed with a known standard deviation of $8$ g. A random sample of $25$ bags has a mean mass of $1012$ g.

   (a)

   What is the standard error of the sample mean?

   [1 mark]

   • A$8$
   • B$0.32$
   • C$1.6$
   • D$40$

   (b)

   Which is the $95\%$ confidence interval for the mean mass of a bag?

   [1 mark]

   • A$(996.3,\ 1027.7)$
   • B$(1007.9,\ 1016.1)$
   • C$(1010.4,\ 1013.6)$
   • D$(1008.9,\ 1015.1)$

   (c)

   Find a $90\%$ confidence interval for the mean mass of a bag.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A $95\%$ confidence interval for the mean $\mu$ of a Normal population with known standard deviation $10$ is $(49.2,\ 54.8)$. It was calculated from a random sample.

   (a)

   Which statement is the correct interpretation of this interval?

   [1 mark]

   • AIf the sampling were repeated many times, about $95\%$ of the intervals constructed would contain $\mu$
   • B$95\%$ of the individual values in the population lie between $49.2$ and $54.8$
   • CThe sample mean has a $95\%$ chance of lying between $49.2$ and $54.8$
   • DThe population mean is certain to lie between $49.2$ and $54.8$

   (b)

   What was the size of the sample?

   [1 mark]

   • A$35$
   • B$49$
   • C$7$
   • D$13$

   (c)

   Use the confidence interval to carry out a test of $H_0:\mu=50$ against $H_1:\mu\neq50$ at the $5\%$ significance level, stating your conclusion.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time taken, in minutes, for a train journey is Normally distributed with a known standard deviation of $4.5$ minutes. A random sample of $36$ journeys has a mean time of $52.4$ minutes.

   (a)

   Find a $99\%$ confidence interval for the mean journey time.

   [3 marks]

   (b)

   A $95\%$ confidence interval for the mean journey time is to have a width of at most $2$ minutes. Find the smallest sample size needed.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lengths of a species of fish are Normally distributed with a known standard deviation of $3.2$ cm. A biologist measures a random sample of $64$ fish and finds a mean length of $21.7$ cm. A fisheries manager claims that the mean length of the species is $23$ cm.

   (a)

   Find a $95\%$ confidence interval for the mean length of the fish, and use it to comment on the manager's claim.

   [6 marks]

   (b)

   A second random sample of $100$ fish has a mean length of $22.1$ cm.
   (i) Find a $99\%$ confidence interval for the mean length from this sample.
   (ii) A colleague says: 'The two samples gave different intervals, so at least one of the intervals must be wrong.' Evaluate this statement.

   [6 marks]

   Total for question 4: 12 marks

End of questions