One-sample t-testAQA A-Level Further Maths: Subtopic test

1. 1
   A café claims that the mean volume of its large coffees is $250$ ml. Volumes are normally distributed. A customer measures $9$ randomly chosen large coffees and finds a sample mean of $243.4$ ml and an unbiased estimate of the population variance of $81$ ml$^2$.

   (a)

   How many degrees of freedom does the $t$-statistic have in a test of the mean using this sample?

   [1 mark]

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

   (b)

   Calculate the value of the test statistic for a test of $H_0:\mu=250$.

   [1 mark]

   • A$-2.2$
   • B$-0.73$
   • C$-0.24$
   • D$2.2$

   (c)

   A test is carried out at the $5\%$ significance level of $H_0:\mu=250$ against $H_1:\mu<250$. The critical value of $t$ is $-1.860$. State the conclusion of the test, in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A gardener claims that a variety of sunflower grows to a mean height of $180$ cm. Heights are normally distributed. A random sample of $10$ plants has heights $x$ cm with $\sum x=1840$ and $\sum x^2=339\,136$.

   (a)

   Find an unbiased estimate of the population variance.

   [1 mark]

   • A$57.6$ cm$^2$
   • B$64$ cm$^2$
   • C$8$ cm$^2$
   • D$576$ cm$^2$

   (b)

   Calculate the value of the test statistic for a test of $H_0:\mu=180$.

   [1 mark]

   • A$1.50$
   • B$0.50$
   • C$5.00$
   • D$1.58$

   (c)

   A test is carried out at the $5\%$ significance level of $H_0:\mu=180$ against $H_1:\mu\neq180$. The critical values of $t$ are $\pm2.262$. State the conclusion of the test, in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   An environment agency states that the mean nitrate concentration in a river is $25$ mg per litre. Concentrations are normally distributed. Residents suspect that the mean is higher, and the agency takes $6$ random readings, in mg per litre: $27.4,\ 25.9,\ 28.1,\ 26.5,\ 29.0,\ 27.7$.

   (a)

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

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether the mean nitrate concentration is higher than $25$ mg per litre. The critical value of $t$ for this test is $2.015$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bakery's machine is set to make loaves with mean mass $800$ g. Masses are normally distributed. After the machine is serviced, an inspector weighs a random sample of $12$ loaves and finds a sample mean of $792.5$ g and an unbiased estimate of the population variance of $144$ g$^2$.

   (a)

   (i) Test, at the $5\%$ significance level, whether the mean mass of loaves has changed. The critical values of $t$ are $\pm2.201$.
   (ii) State an assumption needed for your test.

   [6 marks]

   (b)

   Before weighing, the inspector suspected that the machine now makes lighter loaves, and so decides to test $H_0:\mu=800$ against $H_1:\mu<800$ at the $5\%$ level. The critical value of $t$ is $-1.796$.
   (i) Carry out this test.
   (ii) Explain why the conclusion differs from that in (a), and evaluate which conclusion the inspector should report.

   [6 marks]

   Total for question 4: 12 marks

End of questions