One-sample t-testAQA A-Level Further Maths: Mind map

Exam questions on One-sample t-test

1. A café claims that the mean volume of its large coffees is $250$ ml. Volumes are normally distributed. A customer measures $9$ randomly chosen large coffees and finds a sample mean of $243.4$ ml and an unbiased estimate of the population variance of $81$ ml$^2$.
   A test is carried out at the $5\%$ significance level of $H_0:\mu=250$ against $H_1:\mu<250$. The critical value of $t$ is $-1.860$. State the conclusion of the test, in context.2 marks
2. A gardener claims that a variety of sunflower grows to a mean height of $180$ cm. Heights are normally distributed. A random sample of $10$ plants has heights $x$ cm with $\sum x=1840$ and $\sum x^2=339\,136$.
   A test is carried out at the $5\%$ significance level of $H_0:\mu=180$ against $H_1:\mu\neq180$. The critical values of $t$ are $\pm2.262$. State the conclusion of the test, in context.2 marks
3. An environment agency states that the mean nitrate concentration in a river is $25$ mg per litre. Concentrations are normally distributed. Residents suspect that the mean is higher, and the agency takes $6$ random readings, in mg per litre: $27.4,\ 25.9,\ 28.1,\ 26.5,\ 29.0,\ 27.7$.
   Calculate the sample mean and an unbiased estimate of the population variance.3 marks