Interpret correlation strength/direction

Edexcel S1 2021 January Q5

A company director wants to introduce a performance-related pay structure for her managers. A random sample of 15 managers is taken and the annual salary, $y$ in $\pounds 1000$, was recorded for each manager. The director then calculated a performance score, $x$, for each of these managers.
The results are shown on the scatter diagram in Figure 1 on the next page.
1. Describe the correlation between performance score and annual salary.
The results are also summarised in the following statistics. $$\sum x = 465 \quad \sum y = 562 \quad \mathrm {~S} _ { x x } = 2492 \quad \sum y ^ { 2 } = 23140 \quad \sum x y = 19428$$
1. Show that $\mathrm { S } _ { x y } = 2006$
2. Find $\mathrm { S } _ { y y }$
Find the product moment correlation coefficient between performance score and annual salary. The director believes that there is a linear relationship between performance score and annual salary.
State, giving a reason, whether or not these data are consistent with the director's belief.
Calculate the equation of the regression line of $y$ on $x$, in the form $y = a + b x$ Give the value of $a$ and the value of $b$ to 3 significant figures.
Give an interpretation of the value of $b$.
Plot your regression line on the scatter diagram in Figure 1 The director hears that one of the managers in the sample seems to be underperforming.
On the scatter diagram, circle the point that best identifies this manager. The director decides to use this regression line for the new performance related pay structure.
1. Estimate, to 3 significant figures, the new salary of a manager with a performance score of 30 \begin{figure}[h]
  \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-15_1390_1408_299_187} \captionsetup{labelformat=empty} \caption{Figure 1}
  \end{figure} \includegraphics[max width=\textwidth, alt={}, center]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_2654_99_115_9} Annual salary (£1000) \begin{figure}[h]
  \captionsetup{labelformat=empty} \caption{Only use this scatter diagram if you need to redraw your line.} \includegraphics[alt={},max width=\textwidth]{4f034b9a-94c8-42f2-bd77-9adec277aba6-17_1378_1143_402_468}
  \end{figure}

OCR S1 2012 June Q1

1 For each of the last five years the number of tourists, $x$ thousands, visiting Sackton, and the average weekly sales, $\pounds y$ thousands, in Sackton Stores were noted. The table shows the results.

Year	2007	2008	2009	2010	2011
$x$	250	270	264	290	292
$y$	4.2	3.7	3.2	3.5	3.0

Calculate the product moment correlation coefficient $r$ between $x$ and $y$.
It is required to estimate the average weekly sales at Sackton Stores in a year when the number of tourists is 280000 . Calculate the equation of an appropriate regression line, and use it to find this estimate.
Over a longer period the value of $r$ is - 0.8 . The mayor says, "This shows that having more tourists causes sales at Sackton Stores to decrease." Give a reason why this statement is not correct.

WJEC Further Unit 2 2018 June Q7

7. A university professor conducted some research into factors that affect job satisfaction. The four factors considered were Interesting work, Good wages, Job security and Appreciation of work done. The professor interviewed workers at 14 different companies and asked them to rate their companies on each of the factors. The workers' ratings were averaged to give each company a score out of 5 on each factor. Each company was also given a score out of 100 for Job satisfaction.
The following graph shows the part of the research concerning Job Satisfaction versus Interesting work. \begin{figure}[h]

\captionsetup{labelformat=empty} \caption{Job satisfaction versus Interesting work} \includegraphics[alt={},max width=\textwidth]{3cf25b54-d4a1-4d30-b632-3f6d3182a930-6_647_1330_959_340}

\end{figure}

Calculate the equation of the least squares regression line of Job satisfaction ( $y$ ) on Interesting work ( $x$ ), given the following summary statistics. $$\begin{array} { l l l } \sum x = 46 \cdot 2 , & \sum y = 898 , & S _ { x x } = 3 \cdot 48
S _ { x y } = 49 \cdot 45 , & S _ { y y } = 1437 \cdot 714 , & n = 14 \end{array}$$
Give two reasons why it would be inappropriate for the professor to use this equation to calculate the score for Interesting work from a Job satisfaction score of 90.