The equation for the line of good fit is given by P = - 4 t + 116, where t is time.
Over time, is the value of P constant, increasing, or decreasing?
The equation d = - 0.66 h + 55 represents the line of good fit relating the air humidity, h, and the depth, d metres, of snow in an area.
Use the equation to determine the snow depth when the air humidity is 0.6.
Solve for h, the level of air humidity you would expect to achieve a snow depth of 54.472 \text{ m}.
The average monthly temperature and the average wind speed, in knots, in a particular location was plotted over several months. The graph shows the points for each month’s data and their line of good fit:
Use the line of good fit to approximate the wind speed on a day when the temperature is 5\degree \text{C}.
Determine whether the following graphs show a negative relationship between the two variables:
The following scatter plot graphs data for the number of people in a room and the room temperature collected by a researcher.
Sketch the line of good fit for this data.
Use your line of good fit to estimate the room temperature when there are:
35 people in the room
100 people in the room
The following scatter plot shows the data for two variables, x and y:
Sketch the line of good fit for this data.
Use your line of good fit to estimate the value of y when:
x = 4.5
x = 9
The following scatter plot graphs data for the number of copies of a particular book sold at various prices:
Sketch the line of good fit for this data.
Use your line of good fit to find the number of books that will be sold when the price is:
\$33
\$18
The following scatter plot graphs data for the number of copies of a particular book sold at various prices.
Sketch the line of good fit for this data.
Use the line of good fit to find the number of books that will be sold when the price is:
\$33
\$18
Is the relationship between the price of the book and the number of copies sold positive or negative?
The following scatter plot graphs data for the number of ice blocks sold at a shop on days with different temperatures.
Sketch the line of good fit for this data.
Use your answer line of good fit to estimate the number of ice blocks that will be sold on a:
31 \degree \text{C} day
42 \degree \text{C} day
Does the number of ice blocks sold increase or decrease as the temperature increases?
The following scatter plot graphs data for the number of balls hit and the number of runs scored by a batsman:
Sketch the line of good fit for this data.
Use the line of good fit to estimate the number of runs scored by the batsman after hitting:
27 balls
66 balls
Is the relationship between the two variables positive or negative?
Consider the following scatter plot:
Is the relationship between the x and y variables positive or negative?
Sketch the line of good fit for this data.
Which of the following could be the equation for the line of good fit:
Consider the following scatter plot:
Is the relationship between the x and y variables positive or negative?
Which of the following could be the equation for the line of good fit:
Find the equation of the line of good fit shown:
The distance of several locations from the equator and their temperature on a particular day is measured. The values are presented on the following scatter plot:
Determine whether the following could be the equation of the line relating distance \left(x\right) and temperature \left(y\right):
Estimate the distance from the equator, x, if the temperature is 30.59 \degree C.
The depth a diver has descended below the surface of the water is plotted against her lung capacity:
Does the line of good fit have a positive or negative slope?
Find the equation of line of good fit.
Use the line of good fit to estimate the lung capacity, y, at a depth of 4 \text{ m}.
Consider the scatter plot shown:
Find the equation of the line of good fit.
Use the line of good fit to approximate the value of y for x = 6.9.
Find the equation of the line of good fit on the scatter plot shown:
A number of people were asked how many hours each week they spend on the internet. Their results were graphed against their age in the scatter plot and a strong negative correlation was observed. A line of good fit has been drawn:
Determine the x and y-intercepts of the line of good fit.
Using the intercepts, what is the slope of the line of good fit?
Find the equation for the line of good fit.
Consider the outlier on the graph. According to the line of good fit, what should their usage be?
A study into world health wanted to identify the factors that impact life expectancy among countries. In the study, they chose a few countries and measured the amount spent on healthcare per person per year against the average life expectancy in that country. They discovered a strong positive correlation between the amount spent on healthcare and the life expectancy in the country, and wanted to determine a line of good fit for the data. The table shows the data from the five countries:
| \text{Healthcare per person } (\$A) | \text{Life expectancy } (L) |
|---|---|
| 100 | 27.5 |
| 400 | 44 |
| 1100 | 52.5 |
| 1500 | 71.5 |
| 2100 | 74.5 |
To find one point on the line of good fit, they took the average of the values they had for each variable. Determine the coordinates of this point using the table of values.
Another point that was determined on the line of good fit is \left(700, 45.5\right). Find the slope of the line of good fit.
Determine the value of b, the vertical intercept of the line.
Hence determine the equation of the line of good fit that relates A and L.
A country is currently spending \$30 on healthcare per person each year. According to the line of good fit, by how much would the life expectancy of the country increase if healthcare spending is increased to \$57 per person each year?
The table shows the number of people who went to watch a movie x weeks after it was released:
| \text{Weeks }(x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| \text{Number of people } (y) | 17 | 17 | 13 | 13 | 9 | 9 | 5 |
Plot the points from the table on a number plane.
Which of the following equations could be the line of good fit:
Graph this line of good fit.
Use this equation of the line of good fit to find the number of people who went to watch the movie 10 weeks after it was released.
The table shows data on the number of kilograms of litter collected each week in a national park x weeks after the park managers started an anti-littering campaign:
| \text{Weeks }(x) | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| \text{Kilograms of litter collected } (y) | 2.9 | 2.5 | 2.5 | 2.3 | 2.1 | 1.9 | 1.7 |
Plot the points from the table on a number plane.
Which of the following equations could be the line of good fit:
Graph the line of good fit.
Use the equation of the line of good fit to find the number of kilograms of litter collected 12 weeks after the start of the anti-littering campaign.
Consider the following table showing the number of eggs lain versus the number of ducks:
| Ducks | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Eggs | 4 | 11 | 14 | 21 | 24 | 31 | 34 | 41 |
Plot the points from the table on a number plane.
Sketch a line of good fit for the data.
Find the slope of the line of good fit, given that the line passes through \left(5, 25\right) and \left( - 1 , - 5 \right).
Find the y-intercept of the line of good fit.
Use the equation to find the number of eggs laid by 30 ducks.
The following table shows the temperature (\degree\text{C} ) of a cooling metal versus time:
| Minutes | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| Temperature | 27 | 27 | 23 | 23 | 19 | 19 |
Plot the points from the table on a number plane.
Sketch a line of good fit for the data.
Find the slope of the line of good fit, given that the line passes through \left(5, 20\right) and \left(3, 24\right).
Find the y-intercept of the line of good fit.
Use the equation to find the number of minutes required to reach the temperature of 15 \degree C in terms of x.
Determine if the following predictions are extrapolations or interpolations:
A prediction for the y-value when x = 5 is made from the data set below:
| x | 4 | 7 | 8 | 11 | 12 | 13 | 17 | 18 | 19 | 20 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 0 | 2 | 4 | 7 | 6 | 4 | 8 | 8 | 11 | 8 |
A prediction for the y-value when x = 33 is made from the data set below:
| x | 37 | 54 | 58 | 59 | 43 | 55 | 60 | 38 | 64 | 35 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 72 | 53 | 26 | 21 | 73 | 47 | 12 | 102 | 10 | 112 |
A prediction of y = 95.69 is made from the data set using the line of good fit \\\\\ y = - 0.07 x + 96.18 for the data below:
| x | 19 | 10 | 1 | 7 | 14 | 11 | 2 | 5 | 17 | 8 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 94 | 94.4 | 97 | 96.4 | 94.4 | 97.8 | 94.9 | 95.9 | 96 | 94.4 |
A prediction of y = 72.77 is made from the data set using the line of good fit \\ y = 1.26 x - 57.01 for the data below:
| x | 93 | 57 | 86 | 97 | 78 | 96 | 68 | 69 | 54 | 92 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 51.2 | 25.4 | 38.9 | 58.6 | 38.2 | 60.8 | 26.3 | 28.5 | 5.4 | 92 |
The data below has a correlation coefficient r = 0.93 and a line of good fit y = 0.84 x + 2.66:
| x | 10 | 7 | 14 | 1 | 6 | 13 | 19 | 5 | 20 | 9 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 9 | 10 | 14 | 3 | 6 | 10 | 19 | 9 | 21 | 13 |
Predict the value of y when x = 15.
Comment on the reliability of the prediction in part (a), giving reasons.
The data below has a correlation coefficient r = 0.46 and a line of good fit y = 0.64 x + 54.31:
| x | 37 | 33 | 52 | 100 | 65 | 81 | 83 | 18 | 59 | 51 |
|---|---|---|---|---|---|---|---|---|---|---|
| y | 47.9 | 36.4 | 145.6 | 137 | 74 | 93.8 | 78.4 | 82.4 | 101.2 | 117.3 |
Predict the value of y when x = 49.
Comment on the reliability of the prediction in part (a), giving reasons.
A car company looked at the relationship between how much it had spent on advertising, A, and the amount of sales (in thousands of dollars) each month, S, over several months. The data has been plotted on the scatter graph and a line of good fit drawn. Two points on the line are \left(2000, 300\right) and \left(3500, 450\right).
Using the two given points, find the slope of the line of good fit.
Find the equation of the line of good fit.
Use the line of good fit to estimate the number of sales next month if \$3900 is to be spent on advertising.
Comment on the reliability of the prediction in part (c), giving reasons.
Several cars underwent a brake test and their age was measured against their stopping distance (in metres). The scatter plot shows the results and a line of good fit that approximates the relationship:
According to the line, what is the stopping distance of a car that is 6 years old?
The line of good fit goes through two of the points as shown. Find the slope of the line of good fit.
Find the equation of the line of good fit.
Use the line of good fit to estimate the stopping distance of a car that is 5.5 years old.
Comment on the reliability of the prediction in part (c), giving reasons.
A line of good fit has been drawn to approximate the relationship between sea temperature (\degree\text{C}), T, and the area of healthy coral, A (in hectares), in a particular location. Two particular points, \left(1, 435\right) and \left(23, 105\right), lie on the line.
Calculate the slope of the line.
Find the equation of the line of good fit in the form A = m T + c.
Using the line of good fit, predict the amount of healthy coral A in hectares when the sea temperature is 30 \degree C.
Comment on the reliability of the prediction in part (c), giving reasons.
In a laboratory experiment, a scientist measures the time T (in seconds) that it takes for a chemical reaction to finish, after adding acids of different strengths P (measured as pH). The pH scale, which is used to measure the strength of an acid, allows values from -7 (a strong acid) to +7 (a very strong base). The results are shown in the following table:
| pH | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 |
|---|---|---|---|---|---|---|---|---|---|
| Reaction times | 187 | 176 | 137 | 155 | 153 | 125 | 103 | 124 | 101 |
The least-squares equation for the given data is T = - 10 P + 140.
Use the least-squares equation to predict the reaction time when the acidity is 7 pH.
Use the least-squares equation predict the reaction time when the acidity is 1.5 pH.
Comment on the reliability of the prediction in part (a), giving reasons.
Comment on the reliability of the prediction in part (b), giving reasons.
Soil salinity is a problem that affects large areas of farmland in Australia. A farmer has measured wheat production W (in tonnes per hectare) for a number of paddocks with various salt levels S (in kilograms per hectare).
The data collected is shown in the scatter plot below with the least-squares equation \\ W = - \dfrac{3}{5} S + 300:
Use the least-squares equation predict the wheat production W (in tonnes per hectare) if the salt levels of the paddock are 200 \text{ kg} per hectare.
Use the least-squares equation predict the wheat production W (in tonnes per hectare) if the salt levels of the paddock are 600\text{ kg} per hectare.
Comment on the reliability of the prediction in part (a), giving reasons.
Comment on the reliability of the prediction in part (b), giving reasons.
The fleet manager for the Australian Automotive Association wants to estimate how car maintenance costs C (in hundreds of dollars) are related to the distance K (in thousands of kilometres) driven each year.
The data collected is shown in the scatter plot below with the least-squares equation \\ C = 9.9 K + 146.2:
Use the least-squares equation to predict the annual maintenance cost for a car that is driven 40\,000\text{ km} per year.
Use the least-squares equation to predict the annual maintenance cost for a car that is driven 10\,000\text{ km} per year.
Comment on the reliability of the prediction in part (a), giving reasons.
Comment on the reliability of the prediction in part (b), giving reasons.