Further Statistics 2: Confidence intervals and the t-distributionAQA A-Level Further Maths: Topic test

1. 1
   The lengths of steel rods made by a machine are normally distributed with standard deviation $0.8$ mm. A random sample of $40$ rods has mean length $150.3$ mm. The machine is meant to produce rods of mean length $150.0$ mm.

   (a)

   Find the standard deviation of the sample mean, to $3$ significant figures.

   [1 mark]

   • A$0.020$ mm
   • B$0.101$ mm
   • C$0.126$ mm
   • D$5.06$ mm

   (b)

   Which value is the lower limit of a $95\%$ confidence interval for the population mean?

   [1 mark]

   • A$150.05$ mm
   • B$149.97$ mm
   • C$150.09$ mm
   • D$148.73$ mm

   (c)

   Use the $95\%$ confidence interval to comment on whether the machine is producing rods with mean length $150.0$ mm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The mass of compost in bags sold by a garden centre is normally distributed. A random sample of $9$ bags has sample mean $20.4$ kg and an unbiased estimate of the population variance of $1.44$ kg$^2$. A $95\%$ confidence interval for the population mean is to be found.

   (a)

   How many degrees of freedom does the relevant $t$-distribution have?

   [1 mark]

   • A$9$
   • B$8$
   • C$7$
   • D$10$

   (b)

   Which is the critical value of $t$ to use?

   [1 mark]

   • A$2.262$
   • B$1.860$
   • C$1.960$
   • D$2.306$

   (c)

   Explain why the $t$-distribution is used rather than the normal distribution, and state the assumption about the population that this requires.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The times taken by a restaurant to deliver orders are normally distributed. A random sample of $100$ delivery times, in minutes, has mean $52.4$ and standard deviation $s=6.5$.

   (a)

   Construct a $98\%$ confidence interval for the mean delivery time. Give the limits to $3$ significant figures.

   [3 marks]

   (b)

   A second study assumes the population standard deviation is exactly $6.5$ minutes. Find the least sample size for which a $95\%$ confidence interval for the mean has width at most $2$ minutes.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A car maker claims that the mean fuel consumption of a new model is $5.2$ litres per $100$ km. Fuel consumption is normally distributed. A random sample of $8$ cars gives these values, in litres per $100$ km: $5.6,\ 5.9,\ 5.1,\ 5.7,\ 5.4,\ 6.0,\ 5.5,\ 5.8$. These data have $\sum x^2=253.72$. A consumer group suspects that the mean is higher than the claim.

   (a)

   Test the consumer group's suspicion at the $5\%$ significance level.

   [6 marks]

   (b)

   Construct a $99\%$ confidence interval for the mean fuel consumption. Hence comment on whether the interval is consistent with your conclusion in part (a).

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The yield of apple trees in an orchard is normally distributed with standard deviation $3.0$ kg. A $95\%$ confidence interval for the mean yield, based on a random sample of $n$ trees, is $(17.16,\ 18.84)$ kg.

   (a)

   What is the sample mean?

   [1 mark]

   • A$18.00$ kg
   • B$17.50$ kg
   • C$18.84$ kg
   • D$1.68$ kg

   (b)

   What is the value of $n$?

   [1 mark]

   • A$7$
   • B$12$
   • C$16$
   • D$49$

   (c)

   A $90\%$ confidence interval is constructed from the same sample. State whether it is narrower or wider than the $95\%$ interval, giving a reason.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A school's records say that pupils take $14.0$ minutes on average to walk to school. A teacher thinks this is no longer true. Walking times are normally distributed. A random sample of $10$ pupils has sample mean $12.3$ minutes and unbiased estimate of the population standard deviation $2.1$ minutes.

   (a)

   Which pair of hypotheses should be used?

   [1 mark]

   • A$H_0:\mu=14.0$ and $H_1:\mu<14.0$
   • B$H_0:\bar{x}=14.0$ and $H_1:\bar{x}\ne14.0$
   • C$H_0:\mu=14.0$ and $H_1:\mu\ne14.0$
   • D$H_0:\mu\ne14.0$ and $H_1:\mu=14.0$

   (b)

   Calculate the value of the test statistic, to $3$ significant figures.

   [1 mark]

   • A$-0.810$
   • B$-2.56$
   • C$-8.10$
   • D$-2.43$

   (c)

   The critical values for a two-tailed test at the $5\%$ significance level are required. Use them to complete the test and state your conclusion in context.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The vitamin C content of cartons of orange juice, in mg per $100$ ml, is normally distributed. A random sample of $12$ cartons has $\sum x=507.6$ and $\sum x^2=21544.08$.

   (a)

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

   [3 marks]

   (b)

   Construct a $95\%$ confidence interval for the mean vitamin C content.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A hospital records the time, in minutes, that a nurse spends with each patient. Times are normally distributed. A pilot sample of $6$ patients has $\sum x=83.2$ and $\sum x^2=1157.9$. A later random sample of $120$ patients has mean $14.9$ minutes and $s=2.8$ minutes. A manager believes that the mean time is $14.0$ minutes.

   (a)

   Use the pilot sample to construct a $95\%$ confidence interval for the mean time.

   [6 marks]

   (b)

   Construct a $95\%$ confidence interval from the later sample. Give two reasons why it is narrower than your interval in part (a), and state what each interval suggests about the manager's belief.

   [6 marks]

   Total for question 8: 12 marks

End of questions