Describe correlation from scatter diagram

11 A trainer was asked to give a lecture on population profiles in different Local Authorities (LAs) in the UK. Using data from the 2011 census, he created the following scatter diagram for 17 selected LAs. \begin{figure}[h] \end{figure} He selected the 17 LAs using the following method. The proportions of people aged 18 to 24 and aged 65+ in any Local Authority are denoted by \(P _ { \text {young } }\) and \(P _ { \text {senior } }\) respectively. The trainer used a spreadsheet to calculate the value of \(k = \frac { P _ { \text {young } } } { P _ { \text {senior } } }\) for each of the 348 LAs in the UK. He then used specific ranges of values of \(k\) to select the 17 LAs.

1. Estimate the ranges of values of \(k\) that he used to select these 17 LAs.
2. Using the 17 LAs the trainer carried out a hypothesis test with the following hypotheses. \(\mathrm { H } _ { 0 }\) : There is no linear correlation in the population between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\). \(\mathrm { H } _ { 1 }\) : There is negative linear correlation in the population between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\).
   He found that the value of Pearson's product-moment correlation coefficient for the 17 LAs is - 0.797 , correct to 3 significant figures.
   1. Use the table on page 9 to show that this value is significant at the \(1 \%\) level. The trainer concluded that there is evidence of negative linear correlation between \(P _ { \text {young } }\) and \(P _ { \text {senior } }\) in the population.
   2. Use the diagram to comment on the reliability of this conclusion.
3. Describe one outstanding feature of the population in the areas represented by the points in the bottom right hand corner of the diagram.
4. The trainer's audience included representatives from several universities. Suggest a reason why the diagram might be of particular interest to these people. \begin{table}[h]
   \captionsetup{labelformat=empty} \caption{Critical values of Pearson's product-moment correlation coefficient}

   \multirow{2}{*}{1-tail test 2-tail test}	5\%	2.5\%	1\%	0.5\%
   	10\%	5\%	2\%	1\%
   \(n\)
   1	-	-	-	-
   2	-	-	-	-
   3	0.9877	0.9969	0.9995	0.9999
   4	0.9000	0.9500	0.9800	0.9900
   5	0.8054	0.8783	0.9343	0.9587
   6	0.7293	0.8114	0.8822	0.9172
   7	0.6694	0.7545	0.8329	0.8745
   8	0.6215	0.7067	0.7887	0.8343
   9	0.5822	0.6664	0.7498	0.7977
   10	0.5494	0.6319	0.7155	0.7646
   11	0.5214	0.6021	0.6851	0.7348
   12	0.4973	0.5760	0.6581	0.7079
   13	0.4762	0.5529	0.6339	0.6835
   14	0.4575	0.5324	0.6120	0.6614
   15	0.4409	0.5140	0.5923	0.6411
   16	0.4259	0.4973	0.5742	0.6226
   17	0.4124	0.4821	0.5577	0.6055
   18	0.4000	0.4683	0.5425	0.5897
   19	0.3887	0.4555	0.5285	0.5751
   20	0.3783	0.4438	0.5155	0.5614
   21	0.3687	0.4329	0.5034	0.5487
   22	0.3598	0.4227	0.4921	0.5368
   23	0.3515	0.4132	0.4815	0.5256
   24	0.3438	0.4044	0.4716	0.5151
   25	0.3365	0.3961	0.4622	0.5052
   26	0.3297	0.3882	0.4534	0.4958
   27	0.3233	0.3809	0.4451	0.4869
   28	0.3172	0.3739	0.4372	0.4785
   29	0.3115	0.3673	0.4297	0.4705
   30	0.3061	0.3610	0.4226	0.4629

   \end{table} Turn over for questions 12 and 13

1. A random sample of 15 days is taken from the large data set for Perth in June and July 1987. The scatter diagram in Figure 1 displays the values of two of the variables for these 15 days.

\begin{figure}[h] \end{figure}

1. Describe the correlation. The variable on the \(x\)-axis is Daily Mean Temperature measured in \({ } ^ { \circ } \mathrm { C }\).
2. Using your knowledge of the large data set,
   1. suggest which variable is on the \(y\)-axis,
   2. state the units that are used in the large data set for this variable. Stav believes that there is a correlation between Daily Total Sunshine and Daily Maximum Relative Humidity at Heathrow. He calculates the product moment correlation coefficient between these two variables for a random sample of 30 days and obtains \(r = - 0.377\)
3. Carry out a suitable test to investigate Stav's belief at a \(5 \%\) level of significance. State clearly
   • your hypotheses
   • your critical value
   On a random day at Heathrow the Daily Maximum Relative Humidity was 97\%
4. Comment on the number of hours of sunshine you would expect on that day, giving a reason for your answer.

1. Marc took a random sample of 16 students from a school and for each student recorded

• the number of letters, \(x\), in their last name
• the number of letters, \(y\), in their first name

His results are shown in the scatter diagram on the next page.

1. Describe the correlation between \(x\) and \(y\). Marc suggests that parents with long last names tend to give their children shorter first names.
2. Using the scatter diagram comment on Marc's suggestion, giving a reason for your answer. The results from Marc's random sample of 16 observations are given in the table below.

   \(x\)	3	6	8	7	5	3	11	3	4	5	4	9	7	10	6	6
   \(y\)	7	7	4	4	6	8	5	5	8	4	7	4	5	5	6	3

3. Use your calculator to find the product moment correlation coefficient between \(x\) and \(y\) for these data.
4. Test whether or not there is evidence of a negative correlation between the number of letters in the last name and the number of letters in the first name. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance
   \section*{Question 2 continued.}
   \includegraphics[max width=\textwidth, alt={}]{10736735-3050-43eb-9e76-011ca6fa48b8-05_1125_1337_294_372}
   \section*{Question 2 continued.} \section*{Question 2 continued.}