Problems
You already know each piece separately: covariance, $r$, the two regression lines. Exam problems almost never give you directly what you need; they give you different pieces (two lines, or $r$ and a variance, or just summations) and you have to combine the formulas in the right order. This section is that training.
The formula map and how to travel round it
When studying the statistical relationship between two variables $X$ and $Y$, the following regression lines were obtained: Line of $Y$ on $X$: $y = 0,5x + 4$ Line of…
When studying the statistical relationship between two variables $X$ and $Y$, the following regression lines were obtained:
Line of $Y$ on $X$: $y = 0,5x + 4$
Line of $X$ on $Y$: $x = 1,5y - 3$
Calculate the coordinates of the mean point (centre of gravity) of the bivariate distribution $(\bar{x}, \bar{y})$.
The regression lines of a bivariate distribution are given by the equations $y = -0,5x + 8$ (line of $Y$ on $X$) and $x = -1,28y + 10$ (line of $X$ on $Y$). Calculate…
The regression lines of a bivariate distribution are given by the equations $y = -0,5x + 8$ (line of $Y$ on $X$) and $x = -1,28y + 10$ (line of $X$ on $Y$).
Calculate the linear correlation coefficient $r$ and find the marginal means $\bar{x}$ and $\bar{y}$.
Reconstructing data from summations
A technology company wants to predict the number of sales of a new smartphone ($Y$, in thousands of units) based on daily social media mentions ($X$, in thousands). Data…
A technology company wants to predict the number of sales of a new smartphone ($Y$, in thousands of units) based on daily social media mentions ($X$, in thousands). Data from $8$ previous launches have been collected, obtaining the following summations:
$\sum x_i = 160$, $\sum y_i = 400$, $\sum x_i^2 = 3600$, $\sum y_i^2 = 21500$, $\sum x_i y_i = 8600$.
Calculate the covariance, the linear correlation coefficient, and estimate the sales that will be obtained if $25$ thousand mentions are recorded on the launch day.
The time in hours ($X$) and the distance covered in km ($Y$) by a cyclist over $5$ stages have been measured, obtaining the following pairs of values $(x_i, y_i)$:…
The time in hours ($X$) and the distance covered in km ($Y$) by a cyclist over $5$ stages have been measured, obtaining the following pairs of values $(x_i, y_i)$: $(2, 45)$, $(3, 65)$, $(4, 85)$, $(2, 40)$ and $(5, 110)$. Calculate the covariance of this bivariate distribution.
Changes of variable and affine transformations
In a statistical study on two variables $X$ and $Y$, a change of variables was performed by standardising or rescaling the original data in the form…
In a statistical study on two variables $X$ and $Y$, a change of variables was performed by standardising or rescaling the original data in the form $U = \frac{X - 10}{5}$ and $V = \frac{Y - 20}{2}$. It is known that the regression line of $V$ on $U$ is $v = 0,8u + 1,5$.
By reversing the change of variables, find the equation of the regression line of $Y$ on $X$ in explicit form $y = mx + n$.
The correlation coefficient between monthly income ($X$, in euros) and expenditure on leisure ($Y$, in euros) for a group of people is $r = 0,75$. If we decide to…
The correlation coefficient between monthly income ($X$, in euros) and expenditure on leisure ($Y$, in euros) for a group of people is $r = 0,75$. If we decide to express income in thousands of euros ($X' = \frac{X}{1000}$) and increase leisure expenditure by a fixed amount of $50$ euros due to a subsidy ($Y' = Y + 50$), what will be the new linear correlation coefficient between $X'$ and $Y'$? Justify your answer using the properties of covariance and standard deviations.
Combined covariance of two groups
Two groups of employees from the same multinational company have been studied to analyse the relationship between years of experience ($X$) and monthly salary ($Y$). In…
Two groups of employees from the same multinational company have been studied to analyse the relationship between years of experience ($X$) and monthly salary ($Y$).
In Group A, consisting of $10$ employees, the following were obtained: $\bar{x}_A = 5$, $\bar{y}_A = 1500$, and a covariance $S_{xy,A} = 200$.
In Group B, consisting of $20$ employees, the following were obtained: $\bar{x}_B = 5$, $\bar{y}_B = 1500$, and a covariance $S_{xy,B} = 350$.
Calculate the total covariance $S_{xy}$ if we decided to group all $30$ employees into a single bivariate distribution.
From a sample to a decision: journeys to school
The question is: "Do we need to organise a wider data collection before proposing walking routes?". The population is the students of the school; each student is a unit. The variables are distance to school (km) and journey time (min). Selecting only those who arrive first or who respond voluntarily can bias the results.
For a real investigation, select students at random from all year groups, also record how many do not respond, and work without names or addresses. The sample must allow you to describe the population of interest: five people are not enough to justify a measure for the whole school. We will practise with these fictitious data, not with a real survey:
| Distance (km) | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|
| Time (min) | 12 | 13 | 15 | 14 | 16 |
Reasoned decision: these data are for rehearsing the analysis; before proposing routes we need a larger sample and data on the means of transport and the safety of the routes. Hand in a table, a graph, the parameters with units and a conclusion that states whom it refers to and what cannot be concluded. Compare the conclusion if the sample included only bus users.
Check on the graph how the data lie on both sides of the fitted line (axes truncated):
Working as a team and reviewing errors
In groups, rotate the roles of sample selection, calculation, checking and communication. Before seeing the results, each person writes down their prediction. If two calculations disagree, reproduce one row together and check units and denominators; criticise the method, not the person. If you get stuck, note down which part you understand, split up the task and ask for a specific hint. At the end, each member records one error they corrected, the strategy that helped them and a contribution from another team member that changed their conclusion. Checking your work and perseverance are part of mathematical work.