1 Salbutamol is a drug used to improve lung function. In a medical trial, a random sample of 60 people with impaired lung function was selected. The forced expiratory volume in one second (FEV1) was measured for each person, both before being given salbutamol and again after a two-week course of the drug. The variables \(x\) and \(y\), measured in suitable units, represent FEV1 before and after the two-week course respectively. The data are illustrated in the scatter diagram below, together with the summary statistics for these data.
\includegraphics[max width=\textwidth, alt={}, center]{f3690bc0-3392-4f29-86f7-797d33fab4f1-2_682_1024_502_516}
Summary statistics:
$$n = 60 , \quad \sum x = 43.62 , \quad \sum y = 55.15 , \quad \sum x ^ { 2 } = 32.68 , \quad \sum y ^ { 2 } = 51.44 , \quad \sum x y = 40.66$$
- Calculate the sample product moment correlation coefficient.
- Carry out a hypothesis test at the \(5 \%\) significance level to investigate whether there is positive correlation between FEV1 before and after the course.
- State the distributional assumption which is necessary for this test to be valid. State, with a reason, whether the assumption appears to be valid.
- Explain the meaning of the term 'significance level'.
- Calculate the values of the summary statistics if the data point \(x = 0.55 , y = 1.00\) had been incorrectly recorded as \(x = 1.00 , y = 0.55\).