Hypothesis tests for a Normal meanEdexcel A-Level Maths: Mind map

Exam questions on Hypothesis tests for a Normal mean

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 $\bar X$.
   Given that $P(Z<-2.5)=0.0062$, state the conclusion of the test, in context.2 marks
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 $\bar T$.
   Find the critical values for $\bar T$ for this test.2 marks
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.
   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