Regression line

A study on study habits in Year 12 analyses the relationship between the number of weekly hours spent reading, $X$, and the mark obtained in the Spanish exam, $Y$. The regression line of $Y$ on $X$ is:

$y = 0.65x + 3.2$

1. a) Justify whether the correlation between both variables is positive or negative.
2. b) If a student spends $8$ weekly hours reading, what mark is estimated for them in the exam?

A group of 5th Year students has conducted a study on the relationship between the daily time (in hours) spent using a popular short-video social network ($X$) and the battery percentage remaining on their mobile at the end of the day ($Y$). The data obtained from 6 classmates are as follows:

a) Calculate the centre of gravity of the distribution $(\bar{x}, \bar{y})$.

b) Find the covariance $\sigma_{xy}$ and the linear correlation coefficient $r$. Interpret the result obtained regarding the sign and the strength of the relationship.

c) Determine the equation of the regression line of $Y$ on $X$.

d) If a student uses the social network for 2.5 hours, what battery percentage is estimated to remain? Is this prediction reliable? Justify your answer based on the coefficient of determination $r^2$.

In a study on the performance of a new cryptocurrency, the variables $X$ (advertising expenditure in thousands of euros) and $Y$ (daily transaction volume in millions of euros) have been analysed. After processing the data, the equations of the two regression lines have been obtained:

• Regression line of $Y$ on $X$: $y = 0.85x + 1.24$

• Regression line of $X$ on $Y$: $x = 0.92y - 0.15$

Determine with reasoning:

1. The centre of gravity of the distribution $(\bar{x}, \bar{y})$.

2. Pearson's linear correlation coefficient ($r$).

3. If an advertising expenditure of $15\,000$ € is intended, what would be the expected transaction volume? Justify the reliability of this prediction based on the coefficient of determination.

In a study on sleep habits and academic performance, the following regression lines were obtained for the variables $X$ (daily hours of sleep) and $Y$ (average term mark):

Regression line of $Y$ on $X$: $y = 0.65x + 3.2$

Regression line of $X$ on $Y$: $x = 1.15y - 0.45$

a) Knowing that both lines intersect at the centre of gravity of the distribution, calculate the mean hours of sleep ($\bar{x}$) and the students' average mark ($\bar{y}$).

b) Calculate the linear correlation coefficient $r$ from the gradients of both lines.

c) State with reasoning whether the correlation is positive or negative and whether it is strong or weak.

The grades of 50 students in Economics ($X$) and Geography ($Y$) have a mean of 6.5 in both subjects. The variance in Economics is 4 and in Geography it is 2,25. It is known that the correlation coefficient is $r = 0.8$.

1. a) If a student has obtained a 4 in Economics, estimate their grade in Geography.

1. b) If another student has obtained a 9 in Geography, estimate their grade in Economics using the appropriate regression line.

1. c) Compare both estimates and explain why the same line cannot be used for both cases if we want to minimize the mean squared error.

The charging time of a mobile phone battery ($X$, in minutes) and the percentage of charge reached ($Y$, in %) have been measured. The results are shown in the following graph:

If the equation of the orange line is $y = 1.5x + 5$:

1. a) Estimate the percentage of charge after $60$ minutes.
2. b) Is it a functional or statistical relationship? Justify your answer by observing the position of the points relative to the line.

An athletics coach analyses the relationship between the height of his athletes in cm ($X$) and their stride length in cm ($Y$). The data for 5 athletes are: (160, 110), (170, 120), (175, 125), (180, 132) and (190, 145).

a) Obtain the regression line of $Y$ on $X$.

b) What stride length would be expected for an athlete who is 185 cm tall?

c) Calculate the prediction error (residual) for the athlete who is 180 cm tall. The error is defined as the difference between the actual value and the value estimated by the line: $e = y_i - \hat{y}_i$.

d) If the correlation coefficient were $r = 0.98$, what could you state about the reliability of the coach's predictions?

In a contingency table relating level of education ($X$) and income ($Y$) for a sample, some data have been lost. We know the following parameters:

$\bar{x} = 3$; $\bar{y} = 2\,400$; $s_x^2 = 1.2$; $s_y^2 = 160\,000$ and the regression line of $Y$ on $X$ is $y = 300x + 1\,500$.

1. Calculate the coefficient of determination $R^2$ and interpret the percentage of the variance in income that is explained by the level of education.

2. If an individual has an education level of $X=5$ and their actual income is $3\,200$ €, calculate the prediction error (residual) made by the line.

A courier company knows that the cost of a shipment, $Y$ (in euros), depends on the weight of the parcel, $X$ (in kg), according to the regression line $y = 2.5x + 5$.

If the correlation coefficient is $r = 0.98$, which of the following statements is correct?

1. For every kilogram increase in weight, the cost increases by exactly $2.5$ euros.

2. Predictions made for parcels of a similar weight to those studied will be very reliable.

3. If a parcel weighs $0$ kg, the cost will necessarily be $0$ euros.

4. The relationship between weight and cost is inverse.

A pet adoption centre wants to predict the time it takes for a dog to be adopted ($Y$, in days) based on its age ($X$, in years). They have collected data from 10 dogs and obtained a correlation of $r = 0.85$. The mean age is 4 years and the mean number of days until adoption is 25 days. The standard deviation of age is 2 years and that of the days is 10 days.

a) Find the equation of the regression line.

b) Estimate how long it will take for a 0.5-year-old puppy to be adopted.

c) Estimate how long it will take for a 12-year-old senior dog.

d) Of the two previous estimates, indicate which is more reliable and why, taking into account the range of the original data (ages between 1 and 8 years).

Given the following scatter plot of 12 cars for the variables $X$ (age of a car in years) and $Y$ (market value in thousands of euros), the regression line of $Y$ on $X$, calculated with all 12 points, has been drawn; its equation is $y = -0.88x + 14.59$. Note that there is an outlier at the coordinates $(1, 2)$.

1. a) If we remove that point from the data set, what would you expect to happen to the absolute value of the correlation coefficient? And to the slope of the regression line? Justify your answer.

1. b) Using the given regression line, $y = -0.88x + 14.59$, calculate the estimated value for a 10-year-old car and discuss whether it is appropriate to use this model for cars older than 15 years.