Scatter diagrams：練習問題

A researcher plots a scatter diagram showing the number of hours students spent studying for an exam versus their scores on the exam. The line of best fit is drawn. Which of the following statements is most likely true if the line of best fit has a positive gradient?

A scatter diagram shows a strong negative linear correlation between the age of a car (\(x\) years) and its resale value (\(y\) dollars). A line of best fit is drawn passing through the points \((2, 18000)\) and \((5, 12000)\). Estimate the resale value of a car that is 7 years old.

A set of bivariate data is plotted on a scatter diagram. The mean of the \(x\)-values is \(5.2\) and the mean of the \(y\)-values is \(12.8\). If the line of best fit is drawn by eye, state the coordinates of the point that this line must pass through.

The relationship between the number of workers (\(x\)) and the hours taken to build a wall (\(y\)) is shown on a scatter diagram. It shows that as \(x\) increases, \(y\) decreases. Identify the type of correlation and explain why a linear line of best fit might not be appropriate for very large values of \(x\).

A scatter diagram shows a strong negative correlation between the age of a car, \(a\) years, and its value, \(v\) dollars. The line of best fit is given by the equation \(v = -1500a + 20000\). Predict the value of a car that is 6 years old and explain if this represents an increase or decrease from its initial value.

A researcher investigates the relationship between the age of a tree, \(x\) years, and its height, \(y\) metres. The data for 10 trees is recorded in the table below:

Age (\(x\)): 2, 5, 8, 12, 15, 18, 22, 25, 28, 30
Height (\(y\)): 1.5, 3.4, 4.8, 7.2, 8.5, 10.1, 12.0, 14.2, 15.8, 16.5

(a) Describe the type of correlation shown by this data.
(b) The mean point of the data is \((16.5, 8.9)\). A line of best fit is drawn through this mean point and the point \((5, 3.2)\). Find the equation of this line of best fit in the form \(y = mx + c\).
(c) Use your equation to estimate the height of a tree that is 20 years old.

A researcher collected data on the number of hours spent studying (x) and the score obtained in a mathematics test (y) for 10 students. The data is shown in the table below:

(a) On a graph paper, plot a scatter diagram for this data.

(b) Describe the type of correlation shown in your scatter diagram.

(c) Draw a line of best fit for the data on your scatter diagram.

(d) Use your line of best fit to estimate the test score of a student who studied for 4.5 hours.

(e) Explain whether it is reliable to use this line of best fit to predict the score of a student who studied for 15 hours.