Interpret correlation strength/direction

A question is this type if and only if it asks to describe, interpret, or comment on the type, strength, or direction of correlation from a given correlation coefficient or scatter diagram.

3 questions · Moderate -0.8

5.09c Calculate regression line
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Edexcel S1 2021 January Q5
17 marks Moderate -0.8
  1. 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 \(\mathrm { S } _ { y y }\)
  3. 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.
  4. State, giving a reason, whether or not these data are consistent with the director's belief.
  5. 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.
  6. Give an interpretation of the value of \(b\).
  7. 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.
  8. 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
9 marks Moderate -0.8
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.
Year20072008200920102011
\(x\)250270264290292
\(y\)4.23.73.23.53.0
  1. Calculate the product moment correlation coefficient \(r\) between \(x\) and \(y\).
  2. 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.
  3. 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 Specimen Q4
9 marks Moderate -0.8
A year 12 student wishes to study at a Welsh university. For a randomly chosen year between 2000 and 2017 she collected data for seven universities in Wales from the Complete University Guide website. The data are for the variables: • 'Entry standards' – the average UCAS tariff score of new undergraduate students; • 'Student satisfaction' – a measure of student views of the teaching quality at the university taken from the National Student Survey (maximum 5); • 'Graduate prospects' – a measure of the employability of a university's first degree graduates (maximum 100); • 'Research quality' – a measure of the quality of the research undertaken in the university (maximum 4).
  1. Pearson's product-moment correlation coefficients, for each pairing of the four variables, are shown in the table below. Discuss the correlation between graduate prospects and the other three variables. [2]
    VariableEntry standardsStudent satisfactionGraduate prospectsResearch quality
    Entry standards1
    Student satisfaction-0.0301
    Graduate prospects0.7720.2361
    Research quality0.8660.0660.8271
  2. Calculate the equation of the least squares regression line to predict 'Entry standards'(y) from 'Research quality'(x), given the summary statistics: $$\sum x = 22.24, \sum y = 2522, S_{xx} = 1.0542, S_{xy} = 20193.5, S_{yy} = 122.72.$$ [5]
  3. The data for one of the Welsh universities are missing. This university has a research quality of 3.00. Use your equation to predict the entry standard for this university. [2]