t tests for a mean and the paired t-testEdexcel A-Level Further Maths: Mind map

Exam questions on t tests for a mean and the paired t-test

1. The mass of a chocolate bar is Normally distributed. The label states a mean mass of $50$ g. A random sample of $10$ bars has sample mean $49.2$ g and sample standard deviation $s=2.1$ g. A test is carried out to see whether the mean mass is less than the label states.
   The lower $5\%$ point of $t_9$ is $-1.833$. Complete the test at the $5\%$ significance level and state your conclusion in context.2 marks
2. A random sample of $8$ observations from a Normal population has sample mean $24.3$ and sample standard deviation $s=1.9$. The population variance is unknown.
   Explain why the $t$ distribution is used rather than the Normal distribution, and state the assumption needed about the population.2 marks
3. A supplier claims that the mean length of its rods is $12.0$ cm. The lengths are Normally distributed. A random sample of $8$ rods has $\sum x=99.4$ and $\sum x^2=1236.48$, where $x$ is the length in cm.
   Calculate the sample mean and an unbiased estimate of the population variance.3 marks