Hypothesis tests: correlation and Normal mean – Year 13 A level maths test to download as PDF

A café owner records the daily temperature and the sales of ice lollies over 12 days and finds \(r=0.58\). The 5% two-tailed critical value for \(n=12\) is 0.5760.

1. Write down suitable hypotheses for a test for correlation. [2 marks]
2. Carry out the test and state the conclusion in context. [2 marks]

\(X\sim N(75,15^2)\). A random sample of 25 observations is taken.

1. State the distribution of \(\bar X\). [1 mark]
2. Find \(P(\bar X<72)\). [1 mark]
3. Find \(P(\bar X>80)\). [2 marks]

A drinks company says its bottles contain a mean of 330 ml with standard deviation 3 ml. A trading standards officer takes a random sample of 40 bottles and finds a mean of 328.9 ml.

1. Test at the 5% level whether the mean volume is less than 330 ml. [4 marks]
2. State whether the conclusion changes at the 1% level. [1 mark]

A population is \(N(\mu,6^2)\). A sample of 16 is used to test \(H_0:\mu=80\) against \(H_1:\mu\ne80\) at the 5% level.

1. Find the critical region for \(\bar X\). [3 marks]
2. A sample has mean 83.5. State the conclusion. [1 mark]

For \(n=20\) the 5% critical values are 0.3783 (one-tailed) and 0.4438 (two-tailed). A sample gives \(r=0.41\).

1. State the conclusion of a one-tailed test for positive correlation and of a two-tailed test. [2 marks]
2. Explain why the choice of alternative hypothesis matters. [1 mark]