6
- A researcher is investigating the date of the 'start of spring' at different locations around the country.
A suitable date (measured in days from the start of the year) can be identified by checking, for example, when buds first appear for certain species of trees and plants, but this is time-consuming and expensive. Satellite data, measuring microwave emissions, can alternatively be used to estimate the date that land-based measurements would give.
The researcher chooses a random sample of 12 locations, and obtains land-based measurements for the start of spring date at each location, together with relevant satellite measurements. The scatter diagram in Fig. 6.1 shows the results; the land-based measurements are denoted by \(x\) days and the corresponding values derived from satellite measurements by \(y\) days.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-06_732_1342_781_333}
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\end{figure}
Fig. 6.2 shows part of a spreadsheet used to analyse the data. Some rows of the spreadsheet have been deliberately omitted.
\begin{table}[h]
| 1 | A | B | C | D | E | F |
| 1 | | x | \(\boldsymbol { y }\) | \(\boldsymbol { x } ^ { \mathbf { 2 } }\) | \(\boldsymbol { y } ^ { \mathbf { 2 } }\) | xy |
| 2 | | 90 | 102 | 8100 | 10404 | 9180 |
| 3 | |
| 10 | | | | | | |
| 11 | | | | | | |
| 12 | | 94 | 97 | 8836 | 9409 | 9118 |
| 13 | | 99 | 101 | 9801 | 10201 | 9999 |
| 14 | Sum | 1131 | 1227 | 107783 | 126725 | 116724 |
| 15 | | | | | | |
\captionsetup{labelformat=empty}
\caption{Fig. 6.2}
- Calculate the equation of a regression line suitable for estimating the land-based date of the start of spring from satellite measurements.
- Using this equation, estimate the land-based date of the start of spring for the following dates from satellite measurements.
- 60 days
(iii) Comment on the reliability of each of your estimates. - The researcher is also investigating whether there is any correlation between the average temperature during a month in spring and the total rainfall during that month at a particular location. The average temperatures in degrees Celsius and total rainfall in mm for a random selection, over several years, of 10 spring months at this location are as follows.
| Temperature | 4.2 | 7.1 | 5.6 | 3.5 | 8.6 | 6.5 | 2.7 | 5.9 | 6.7 | 4.1 |
| Rainfall | 18 | 26 | 42 | 76 | 15 | 43 | 84 | 53 | 66 | 36 |
\includegraphics[alt={},max width=\textwidth]{3a89edc4-ac93-4691-ade8-4d4665b55202-07_693_880_1174_338}
\captionsetup{labelformat=empty}
\caption{Fig. 6.3}
\end{figure}
(i) Explain why the researcher might come to the conclusion that a test based on Pearson's product moment correlation coefficient may be valid.
(ii) Find the value of Pearson's product moment correlation coefficient.
(iii) Carry out a test at the \(5 \%\) significance level to investigate whether there is any correlation between temperature and rainfall.