Hypothesis test for the mean of a Normal distributionAQA A-Level Maths: Subtopic test

1. 1
   A machine fills bags with sugar. The mass $X$ g of a bag is Normally distributed with standard deviation $6$ g. The mean mass is supposed to be $250$ g, but the operator suspects that the mean has decreased. The operator takes a random sample of $36$ bags and finds that the sample mean is $247.5$ g. The standard deviation is assumed to be unchanged.

   (a)

   Assuming the mean is $250$ g, the distribution of the sample mean $\bar X$ of $36$ bags is

   [1 mark]

   • A$N(250,6^2)$
   • B$N\left(250,\left(\frac16\right)^2\right)$
   • C$N(247.5,1^2)$
   • D$N(250,1^2)$

   (b)

   Which pair of hypotheses should the operator use?

   [1 mark]

   • A$H_0:\bar x=250$, $H_1:\bar x<250$
   • B$H_0:\mu=250$, $H_1:\mu<250$
   • C$H_0:\mu=250$, $H_1:\mu\ne250$
   • D$H_0:\mu<250$, $H_1:\mu=250$

   (c)

   Assuming that the mean is $250$ g, find the probability of obtaining a sample mean of $247.5$ g or less.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time $T$ minutes that a technician takes to complete a task is Normally distributed with standard deviation $4$. A manager claims that the mean time is $30$ minutes. A random sample of $16$ times has mean $31.8$ minutes. The standard deviation is assumed to be $4$.

   (a)

   The manager tests whether the mean time differs from $30$ minutes at the $5\%$ significance level. The critical region for the sample mean $\bar X$ is

   [1 mark]

   • A$\bar X<28.04$
   • B$\bar X>31.65$
   • C$\bar X<28.04$ or $\bar X>31.96$
   • D$\bar X<26.08$ or $\bar X>33.92$

   (b)

   Which conclusion is correct for the sample mean of $31.8$ at the $5\%$ level?

   [1 mark]

   • A$31.8$ is not in the critical region, so there is insufficient evidence that the mean time differs from $30$ minutes
   • B$31.8$ is not in the critical region, so the mean time is exactly $30$ minutes
   • C$31.8$ is greater than $30$, so there is significant evidence that the mean time is greater than $30$ minutes
   • D$31.8$ is in the critical region, so there is significant evidence that the mean time differs from $30$ minutes

   (c)

   The manager repeats the test at the $10\%$ significance level. Find the $p$-value for this two-tailed test and state, with a reason, whether the conclusion changes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lifetime $L$ hours of a type of battery is Normally distributed with standard deviation $12$. The manufacturer claims that the mean lifetime is $500$ hours. A consumer group believes that the mean lifetime is less than this. It tests a random sample of $25$ batteries and finds a sample mean of $495$ hours.

   (a)

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

   [3 marks]

   (b)

   Find the critical region for $\bar L$ for the same test at the $1\%$ significance level, and state the conclusion for the sample mean $495$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rail company states that the journey time $T$ minutes between two cities is Normally distributed with mean $48$ and standard deviation $5$. Passengers believe that the mean journey time is now longer than $48$ minutes. It is assumed that the standard deviation is unchanged.

   (a)

   A random sample of $40$ journeys has mean time $49.6$ minutes. Test, at the $5\%$ significance level, whether the passengers' belief is supported.

   [6 marks]

   (b)

   A different sample, of only $10$ journeys, also has mean time $49.6$ minutes.
   (i) Carry out the test at the $5\%$ significance level using this sample, finding the $p$-value.
   (ii) State your conclusion in context.
   (iii) Explain why the conclusion differs from that for the sample of $40$ journeys.

   [6 marks]

   Total for question 4: 12 marks

End of questions