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.

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.

For each month of a certain year, a weather station recorded the average rainfall per day, \(x\) mm, and the average amount of sunshine per day, \(y\) hours. The results are summarised below. \(n = 12\), \(\Sigma x = 24.29\), \(\Sigma x^2 = 50.146\), \(\Sigma y = 45.8\), \(\Sigma y^2 = 211.16\), \(\Sigma xy = 88.415\).

1. Find the mean values, \(\bar{x}\) and \(\bar{y}\). [1]
2. Calculate the gradient of the line of regression of \(y\) on \(x\). [2]
3. Use the answers to parts (i) and (ii) to obtain the equation of the line of regression of \(y\) on \(x\). [2]
4. Find the product moment correlation coefficient and comment, in context, on its value. [4]
5. Stating your hypotheses, test at the 1% level of significance whether there is negative correlation between average rainfall per day and average amount of sunshine per day. [4]

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 [4 marks]

A bowling alley manager in the UK is concerned about falling revenues. He collects data from the United States, hoping to use what he finds to revive his business in the UK. He finds data which seem to show correlation between margarine consumption and bowling alley revenue. He attempts to carry out some statistical analysis in order to present his findings to the board of directors. He produces the scatter diagram shown below. \includegraphics{figure_5} The product moment correlation coefficient for these data is \(-0.7617\). He carries out a one-tailed test at the 1\% level of significance and concludes that higher margarine consumption is associated with lower revenue generated by bowling alleys.

1. Show all the working for this test. [5]

The manager also conducts a significance test for bowling alley revenue and fish consumption per person. He produces the computer output, shown below, for the analysis of bowling alley revenue versus fish consumption per person. \# Pearson's product-moment correlation
\# data: revenue and fish
\# t = 3.8303, df = 8, p-value = 0.005215
\# alternative hypothesis: true correlation is not equal to 0
\# sample estimates:
\# correlation
\# 0.802423

1. Comment on the correlation between bowling alley revenue and fish consumption per person and what the board of directors should do in light of the manager's findings in part (a) and part (b). [3]
2. Give one possible reason why the board of directors might not be happy with the manager's analysis. [1]