In this question, you will perform a hypothesis test using statistical theory, rather than bootstrap and randomisation distributions. Use the methods described in section 6.1 of the lecture notes and course textbook.

Make sure you use appropriate notation where necessary.

Data collected from 1000 randomly selected private dwellings in Christchurch found that 781 of them were using a heat pump for heating.

Does this provide enough evidence that more than 75% of private dwellings in Christchurch are using a heat pump for heating?Determine at a 5% significance level (= 0.05).

1. (i)What are the null and alternative hypotheses for the research questionhighlightedabove?
2. (ii)Explainwhatconditionsarenecessaryforthistestandcheckiftheyaremet.
3. (iii)Calculate the test statistic. Include your working.
4. (iv)Calculate the p-value for this test.
5. (v)Use your p-value to make a decision and conclusion for your hypotheses in context.