Paper 2 Section B Practice: Hypothesis Testing

Original practice material

These four scenarios total 41 marks. They are a topic-local practice pack, not a complete Paper 2 Section B.

Answers: Hypothesis Testing Answers.

Question 1: Lower-Tailed Test from Summary Information

A filling machine is set so that the mean amount of liquid in a bottle is $500$ ml. The population standard deviation is known to be $12$ ml. A supervisor suspects that the machine is underfilling bottles.

For a random sample of $36$ bottles, the total amount of liquid is $17859.6$ ml.

(a) Calculate the sample mean. [1]

(b) State suitable hypotheses, defining any symbol you use. [2]

(c) Carry out the test at the $5\%$ significance level using a critical-value method. [5]

(d) State the conclusion in context. Explain why the conclusion does not imply that every bottle is underfilled. [2]

[Total: 10]

Question 2: Two-Tailed Test

A manufacturer claims that the mean lifetime of a battery is $48$ hours. The population standard deviation is known to be $5.4$ hours. A random sample of $64$ batteries has mean lifetime $46.7$ hours.

A quality-control engineer wishes to test whether the population mean lifetime has changed.

(a) Explain why a two-tailed test is appropriate. [1]

(b) State the hypotheses, defining any symbol you use. [2]

(c) Carry out the test at the $1\%$ significance level using critical values. [5]

(d) Find the p-value to 3 significant figures and verify that it is consistent with the decision in part (c). [2]

[Total: 10]

Question 3: Upper-Tailed Test with a Diagram

The established population mean score on a standardised diagnostic test is $72$. After participating in a training programme, a random sample of $49$ students has mean score $74.4$. The population standard deviation of scores is known to be $8$.

A researcher tests whether the population mean score after the programme exceeds $72$.

For a $5\%$ upper-tailed test, the critical region is shown below.

(a) State the null and alternative hypotheses, defining any symbol you use. [2]

(b) Calculate the observed test statistic. [2]

(c) Use the diagram and your answer to part (b) to make a decision and state the conclusion in context. [3]

(d) Explain the role of the known population standard deviation in this test. [2]

(e) State one condition about the sample or population model that is needed for the test to be valid. [1]

[Total: 10]

Question 4: Reverse Hypothesis-Test Boundary

The population mean waiting time at a service counter was previously $120$ seconds. A new queueing system is introduced. The population standard deviation is known to remain $10$ seconds. A lower-tailed $Z$-test is to be carried out using a random sample of $25$ waiting times.

The hypotheses are

$$H_0:\mu =120,H_1:\mu <120.$$

(a) State the critical value for the standard normal test statistic at the $5\%$ significance level. [1]

(b) Find the largest sample mean, to 2 decimal places, for which $H_0$ would be rejected. [4]

(c) If the observed sample mean is $117.0$ seconds, state the decision of the test. [2]

(d) Explain why the boundary in part (b) changes if the sample size is increased while the significance level and population standard deviation remain unchanged. [4]

[Total: 11]