Scatter plots and correlation

Look at the following scatter diagrams shown below. Identify in each case whether the correlation is positive, negative or zero.

Match each of the following pairs of variables with the type of scatter diagram you would expect to obtain (increasing, decreasing or with no defined shape):

1. a) The number of kilometres driven by a car and the fuel remaining in its tank.
2. b) The height of a group of teenagers and their mark in an English exam.
3. c) An athlete's training hours and their personal best in a 100-metre sprint.
4. d) The outdoor temperature in summer and the energy consumption of air conditioning units.

Look at the following scatter diagram representing the relationship between weekly training hours ($X$) and points scored in a season by a group of basketball players ($Y$).

Based on the shape of the scatter diagram, answer:

1. a) Is it a positive or negative relationship?
2. b) How would you describe the strength of the correlation: strong or weak?
3. c) What sign do you expect the covariance $\sigma_{xy}$ to have for these data?

We compare two scatter diagrams shown in the following graphs. Both show a positive relationship between the variables.

1. a) In which of the two graphs are the points closer to a possible regression line?
2. b) Which of the two will have a correlation coefficient $r$ closer to $1$?
3. c) If in graph A the value of $r$ is $1$, what reasonable value (between $0$ and $1$) could you assign to graph B?

The weight ($X$) and height ($Y$) of the players in a basketball team have been measured. The scatter diagram shows a very elongated and narrow shape following an upward trend.

What does a "narrow" diagram indicate regarding the strength of the correlation? What would we call the relationship between weight and height in this case (positive or negative)?

The direction of the relationship between two variables indicates how one varies when the other changes. Determine whether a positive correlation, negative correlation exists, or whether the variables are uncorrelated in the following cases:

1. a) The number of absences from class and the final mark for the course.
2. b) A person's shoe size and their average mark in Mathematics.
3. c) The outdoor temperature in summer and the energy consumption of air conditioning units.
4. d) The age in years of a mobile phone and its resale price on the second-hand market.

The covariance $\sigma_{xy}$ provides information about the direction of joint variation of two variables. Without carrying out complex calculations, look at the following data relating the temperature of a soft drink ($X$, in $^\circ C$) to the time it has been out of the fridge ($Y$, in minutes):

1. a) As time increases, does the temperature tend to increase or decrease?
2. b) Justify, based on your previous answer, whether the covariance will be positive, negative or zero.

Pearson's correlation coefficient, $r$, is a dimensionless measure that varies within a specific range. Of the following values, state which ones are not possible for a correlation coefficient and explain why:

1. a) $r = 0,85$
2. b) $r = -1,2$
3. c) $r = 2,5$
4. d) $r = -0,4$
5. e) $r = 0$

The following table shows the distance covered ($X$, in km) and the time taken ($Y$, in min) by a cyclist on training rides:

1. a) Draw the scatter diagram.
2. b) Sketch freehand the line of best fit for the points (regression line).
3. c) Using the line you have drawn, estimate how long it would take the cyclist to cover 45 km. Do you think this estimation is reliable?

You can do it on paper and send a photo with your answer.

A study has been conducted on the relationship between the number of books read per year ($X$) and the score in a reading comprehension test ($Y$) for a group of students. The resulting scatter diagram is shown below:

1. a) State whether a deterministic (functional) relationship or a statistical relationship exists. Explain the difference between both concepts using this example.
2. b) Classify the correlation according to its sign and its strength (strong, moderate or weak).
3. c) Could we state that "reading more books necessarily causes getting a higher mark"? Explain the difference between correlation and causation.

A biologist measures the height of an exotic plant species ($Y$, in cm) as a function of the amount of nitrogenous fertiliser applied ($X$, in mg). She/He observes that, initially, the height increases rapidly with the fertiliser, but beyond a certain amount, the plant becomes saturated and the height stabilises or even decreases slightly due to toxicity.

1. Analyse the scatter diagram and determine whether a linear regression model would be suitable for describing this relationship across the entire domain of $X$.

2. If the biologist calculates Pearson's correlation coefficient for all the data and obtains a value close to $0$, does this mean that the variables are independent? Justify your answer based on the shape of the scatter diagram.

Explain, giving reasons, why the following data could not be correctly represented using a scatter diagram in the context of bivariate statistics:

In a study on urban demography, population density (inhabitants/$km^2$) is represented on the $X$ axis and the average rent price ($€/m^2$) on the $Y$ axis. The resulting scatter diagram displays a very peculiar structure: two clearly distinct groups of points ("clusters"). Group 1 corresponds to peripheral neighbourhoods and Group 2 to historic centre neighbourhoods.

1. If within each separate group the correlation is positive and strong, but when considering all points together the scatter diagram appears to have no clear trend, what problems would we encounter when attempting to calculate a single Pearson correlation coefficient for the entire city?

2. Explain how the existence of hidden qualitative variables (such as neighbourhood type) can "mislead" the interpretation of a bivariate scatter diagram.

Consider a scatter diagram with a very strong positive linear correlation ($r = 0,98$). Suppose that we transform all the values of variable $X$ using the operation $X' = X + 50$ and all the values of variable $Y$ using $Y' = 3 \cdot Y$.

1. Describe what changes the scatter diagram would undergo in the plane (translation, deformation, change of scale).

2. Reason, without performing calculations, whether the new correlation coefficient $r'$ will be greater than, less than, or equal to the original $r$.

3. Would the reliability of the predictions change if we used the new transformed scatter diagram? Explain why.