Hypothesis tests for a Normal meanEdexcel A-Level Maths: Subtopic test

1. 1
   The masses of packets of cereal, in grams, are Normally distributed with standard deviation 4. The packets are labelled with a mean mass of 500 g. A consumer group suspects that the true mean mass $\mu$ is less than 500 g. It tests this at the 1% significance level using a random sample of 25 packets, with sample mean $\bar X$.

   (a)

   If the null hypothesis $\mu=500$ is true, what is the distribution of $\bar X$?

   [1 mark]

   • A$N(500,16)$
   • B$N(500,0.64)$
   • C$N(500,0.8)$
   • D$N(500,3.2)$

   (b)

   The sample mean is 498 g. Find the value of the test statistic.

   [1 mark]

   • A$-0.5$
   • B$-3.13$
   • C$-2.5$
   • D$-12.5$

   (c)

   Given that $P(Z<-2.5)=0.0062$, state the conclusion of the test, in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time $T$ minutes taken by workers to assemble a component is Normally distributed with standard deviation 1.5. A supervisor claims that the mean time $\mu$ is 12 minutes. A researcher believes that the mean time is different, and tests this at the 5% significance level using a random sample of 36 workers, with sample mean $\bar T$.

   (a)

   Which hypotheses should the researcher use?

   [1 mark]

   • A$H_0:\mu=12,\ H_1:\mu\ne12$
   • B$H_0:\bar t=12,\ H_1:\bar t\ne12$
   • C$H_0:\mu=12,\ H_1:\mu>12$
   • D$H_0:\mu\ne12,\ H_1:\mu=12$

   (b)

   If the null hypothesis is true, what is the standard deviation of $\bar T$?

   [1 mark]

   • A$1.5$
   • B$0.0417$
   • C$0.0625$
   • D$0.25$

   (c)

   Find the critical values for $\bar T$ for this test.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lengths of rods made by a machine, in millimetres, are Normally distributed with standard deviation 0.8. The machine is set to produce rods with mean length 50. After maintenance the manager believes that the mean length $\mu$ has increased. A random sample of 16 rods has mean length 50.45 mm.

   (a)

   State the hypotheses for a test at the 5% significance level, and the distribution of the sample mean $\bar X$ if the null hypothesis is true.

   [3 marks]

   (b)

   Carry out the test and state your conclusion in context.

   [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 12. The manufacturer claims that the mean lifetime is 150 hours. A consumer believes that the mean lifetime $\mu$ is less than this and tests a random sample of 40 batteries, with sample mean $\bar L$.

   (a)

   The sample of 40 batteries has a mean lifetime of 145.2 hours. Test the consumer's belief at the 5% significance level.
   (i) State the hypotheses.
   (ii) Find the test statistic.
   (iii) State your conclusion, in context.

   [6 marks]

   (b)

   The consumer repeats the test at the 1% significance level with the same sample.
   (i) Find the critical region for $\bar L$.
   (ii) State the conclusion of the test, in context.
   (iii) Explain how the critical value would differ if the sample had only 10 batteries.

   [6 marks]

   Total for question 4: 12 marks

End of questions