Hypothesis tests for a meanEdexcel International A Level Further Maths: Subtopic test

1. 1
   A manufacturer claims that the lifetimes of its light bulbs are Normally distributed with mean $1200$ hours and standard deviation $80$ hours. A consumer group believes that the mean lifetime is lower than claimed. A random sample of $16$ bulbs has a mean lifetime of $1165$ hours. Assume that the standard deviation is $80$ hours.

   (a)

   Which pair of hypotheses should the consumer group use?

   [1 mark]

   • A$H_0:\mu=1200,\ H_1:\mu<1200$
   • B$H_0:\bar x=1200,\ H_1:\bar x<1200$
   • C$H_0:\mu=1165,\ H_1:\mu<1165$
   • D$H_0:\mu=1200,\ H_1:\mu\neq1200$

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$-0.4375$
   • B$-1.75$
   • C$-7$
   • D$+1.75$

   (c)

   Carry out the test at the $5\%$ significance level and state your conclusion in context.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A machine fills cartons with orange juice. The volume is Normally distributed with standard deviation $6$ ml, and the mean volume is meant to be $500$ ml. A quality manager takes a random sample of $25$ cartons, which has a mean volume of $502.7$ ml, and tests at the $5\%$ significance level whether the mean volume has changed.

   (a)

   What are the critical values of $z$ for this test?

   [1 mark]

   • A$\pm1.645$
   • B$\pm2.326$
   • C$\pm1.96$
   • D$\pm2.576$

   (b)

   What is the value of the test statistic?

   [1 mark]

   • A$0.45$
   • B$1.875$
   • C$11.25$
   • D$2.25$

   (c)

   Complete the test and state your conclusion in context.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time, in minutes, that a customer waits to be served at a call centre has an unknown, positively skewed distribution with standard deviation $3.5$ minutes. The centre claims that the mean waiting time is $5.8$ minutes, but a consumer group believes it is longer. A random sample of $49$ customers has a mean waiting time of $6.4$ minutes.

   (a)

   Explain why the sample mean can be assumed to be approximately Normally distributed, and state its approximate distribution if the centre's claim is correct.

   [3 marks]

   (b)

   Test, at the $5\%$ significance level, whether the mean waiting time is longer than the centre claims.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A road safety group claims that the mean speed of cars on a stretch of road is $48$ mph. An officer records the speeds, $x$ mph, of $100$ randomly chosen cars and finds $\sum x=4930$ and $\sum x^2=246\,613$. The distribution of speeds is not assumed to be Normal.

   (a)

   Test, at the $5\%$ significance level, whether the mean speed differs from $48$ mph. Use the sample to estimate the population variance.

   [6 marks]

   (b)

   (i) Taking the population standard deviation to be $6$ mph, find the critical region for $\bar X$ for a two-tailed test of $H_0:\mu=48$ at the $5\%$ significance level, using a sample of $100$ cars.
   (ii) A second officer records only $12$ cars and plans to use the same method with the sample variance in place of $\sigma^2$. Explain why this may not be valid.

   [6 marks]

   Total for question 4: 12 marks

End of questions