One-tailed test for negative correlation

5 For a random sample of 12 observations of pairs of values \(( x , y )\), the product moment correlation coefficient is - 0.456 . Test, at the \(5 \%\) significance level, whether there is evidence of negative correlation between the variables.

7. A sports scientist wishes to examine the link between resting pulse and fitness. He records the resting pulse, \(p\), of 20 volunteers and the length of time, \(t\) minutes, that each one can run comfortably at 4 metres per second on a treadmill. The results are summarised by $$\Sigma p = 1176 , \quad \Sigma t = 511 , \quad \Sigma p ^ { 2 } = 70932 , \quad \Sigma t ^ { 2 } = 19213 , \quad \Sigma p t = 27188 .$$

1. Calculate the product moment correlation coefficient for these data.
2. Stating your hypotheses clearly, test at the \(1 \%\) level of significance whether there is evidence of people with a lower resting pulse having a higher level of fitness as measured by the test.
3. State an assumption necessary to carry out the test in part (b) and comment on its validity in this case.
   (2 marks)

3 A student is investigating the link between temperature (in degrees Celsius) and electricity consumption (in Gigawatt-hours) in the country in which he lives. The student has read that there is strong negative correlation between daily mean temperature over the whole country and daily electricity consumption during a year. He wonders if this applies to an individual season. He therefore obtains data on the mean temperature and electricity consumption on ten randomly selected days in the summer. The spreadsheet output below shows the data, together with a scatter diagram to illustrate the data. \includegraphics[max width=\textwidth, alt={}, center]{5be067ff-4668-48d6-8ed2-b8dfa3e678f7-3_798_1593_639_251}

1. Calculate Pearson’s product moment correlation coefficient between daily mean temperature and daily electricity consumption. The student decides to carry out a hypothesis test to investigate whether there is negative correlation between daily mean temperature and daily electricity consumption during the summer.
2. Explain why the student decides to carry out a test based on Pearson's product moment correlation coefficient.
3. Show that the test at the \(5 \%\) significance level does not result in the null hypothesis being rejected.
4. The student concludes that there is no correlation between the variables in the summer months. Comment on the student's conclusion.

16 A medical student believes that, in adults, there is a negative correlation between the amount of nicotine in their blood stream and their energy level. The student collected data from a random sample of 50 adults. The correlation coefficient between the amount of nicotine in their blood stream and their energy level was - 0.45 Carry out a hypothesis test at the \(2.5 \%\) significance level to determine if this sample provides evidence to support the student's belief. For \(n = 50\), the critical value for a one-tailed test at the \(2.5 \%\) level for the population correlation coefficient is 0.2787