Hypothesis tests for a meanEdexcel International A Level Further Maths: Mind map

Exam questions on Hypothesis tests for a mean

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.
   Carry out the test at the $5\%$ significance level and state your conclusion in context.2 marks
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.
   Complete the test and state your conclusion in context.2 marks
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.
   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