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

Exam questions on Hypothesis test for the mean of a Normal distribution

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.
   Assuming that the mean is $250$ g, find the probability of obtaining a sample mean of $247.5$ g or less.2 marks
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 $4$.
   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
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.
   Test, at the $5\%$ significance level, the consumer group's belief.3 marks