Aimathic
Login | English | Deutsch

Grade 8 math worksheets

Build your own math worksheets from 35,000+ problems for grades 3 to 12, from fractions to AP Calculus. Every problem comes with step-by-step solutions.

Informal line of best fit

Learning goals

  • Draws or selects a line that follows the overall trend of a scatterplot.
  • Balances points above and below the line reasonably.
  • Uses the line to make approximate predictions.
  • Explains why the line is a model rather than an exact fit.

Click problems to add them to your worksheet.

Informal line of best fit 1
Classify each prediction from a fit line as interpolation or extrapolation. The observed \(x\)-values range from \(4\) to \(18\). a) \(x=12\) b) \(x=20\) c) \(x=5\)

Hints

- Compare each \(x\)-value with the smallest and largest observed \(x\)-values. - Values inside the observed range use the fit line between observed input values; values beyond that range extend the model outside the data.

Solution

1. \(12\) lies inside the observed range, so a) is interpolation. 2. \(20\) lies outside the observed range, so b) is extrapolation. 3. \(5\) lies inside the observed range, so c) is interpolation.

Answer

a) Interpolation b) Extrapolation c) Interpolation
Informal line of best fit 2
Which line, a) or b), is a more reasonable line of fit for the plotted points? Explain your choice.
Figure for problem 560105

Hints

- Compare where the points lie in relation to each line. - A useful line of fit should follow the center of the overall pattern.

Solution

1. The points lie close to an upward straight pattern. 2. In a), the line runs through the middle of the points; in b), the line lies above nearly all of them. 3. Line a) is the better fit.

Answer

a), because the line runs through the middle of the data instead of staying above the points.
Informal line of best fit 3
What value of \(y\) does the fitted line predict when \(x=4\)?
Figure for problem 560107

Hints

- Start at the given x-value on the horizontal axis. - Read the height of the straight line where that vertical position meets it.

Solution

1. Locate \(x=4\) on the horizontal axis and move up to the fitted line. 2. The line is at \(y=8\) there, so the predicted value is about \(8\).

Answer

About \(8\).
Informal line of best fit 4
The scatter plot shows an approximately linear pattern. Which point would best continue the pattern: \((8, 10)\), \((8, 18)\), \((8, 26)\), or \((8, 32)\)?
Figure for problem 549999

Hints

- Estimate where the main pattern would be at the given \(x\)-value. - Choose the point nearest the continuation of the trend.

Solution

1. As \(x\) increases by about \(1\), the plotted \(y\)-values increase by about \(3\). 2. Extending that pattern to \(x=8\) gives a \(y\)-value near \(26\). 3. The point that best continues the pattern is \((8, 26)\).

Answer

\((8, 26)\).
Informal line of best fit 5
Nora draws one straight line of fit through the scatter plot. Explain why a straight line is not a useful summary of this data set.
Figure for problem 550025

Hints

- Check whether the line would miss different regions in the same direction. - A clear curve should not be forced into one straight trend.

Solution

1. The points follow a clear U-shaped pattern. 2. Any straight line would miss the low middle and the high ends in a systematic way. 3. A straight linear fit is not appropriate for this nonlinear data.

Answer

A straight line is not useful because the data follow a strong U-shaped nonlinear pattern.
Informal line of best fit 6
A line of fit predicts \(y=18\) at one point. Use the displayed line to estimate the corresponding \(x\)-value.
Figure for problem 550027

Hints

- Work backward from the \(y\)-value to the line. - Find the horizontal coordinate of the point on the fitted line.

Solution

1. Find the point on the fit line where \(y=18\). 2. The line follows \(y=3x\). 3. Solve \(18=3x\) to get \(x=6\).

Answer

\(x=6\).
Informal line of best fit 7
The scatter plot has a negative linear association. Which candidate is the most reasonable line of fit: \(y=-x+18\), \(y=-2x+18\), or \(y=-3x+18\)? Explain using the point cloud.
Figure for problem 550034

Hints

- All three lines have negative slope, so compare more than direction. - Estimate the cloud's vertical change over a convenient horizontal run. - Check which candidate would stay near the center of the points across the x-range.

Solution

1. All three candidates decrease, so the sign of the slope alone does not decide the question. 2. The data drop by about \(16\) y-units as \(x\) increases by about \(8\), suggesting a slope near \(-2\). 3. The line \(y=-2x+18\) also runs through the center of the cloud, while \(y=-x+18\) is too shallow and \(y=-3x+18\) is too steep. 4. Therefore, \(y=-2x+18\) is the most reasonable fit.

Answer

\(y=-2x+18\), because its slope and placement best match the center of the negative point cloud.
Informal line of best fit 8
The plotted line is \(y=x+1\). Which data point has the largest vertical difference from the line?
Figure for problem 550035

Hints

- Compare each point’s \(y\)-coordinate with the line’s \(y\)-value at the same \(x\)-coordinate. - Use vertical distance, not the straight-line distance to the graph.

Solution

1. Compute the fitted values: at \(x=1, 2, 3, 4, 5\), the line gives \(2, 3, 4, 5, 6\). 2. The vertical differences for the points are \(0, 1, 1, 5, 1\). 3. The point \((4, 10)\) has the largest vertical difference.

Answer

\((4, 10)\).
Informal line of best fit 9
Which displayed line is the better informal fit, line \(a\) or line \(b\)? Explain.
Figure for problem 550040

Hints

- A useful line should reflect the direction of the association. - The center height alone is not enough when the cloud has a trend.

Solution

1. A horizontal line ignores the downward relationship. 2. Line \(b\) matches both the direction and central location of the cloud. 3. Line \(b\) is the better fit.

Answer

Line \(b\).
Informal line of best fit 10
Hana wants an equation for a drawn fit line. Why is it usually better to choose two clear grid-intersection points on the line instead of two original data points near the line?

Hints

- Distinguish the model line from the observations it summarizes. - Choose coordinates that can be read exactly from the line.

Solution

1. The equation must describe the drawn line itself. 2. Data points near the line may not lie exactly on it. 3. Clear points on the line give an accurate slope and intercept for that line.

Answer

Use clear points on the fit line because nearby data points may not lie on the line and can give the wrong equation.
Informal line of best fit 11
At \(x=9\), which prediction from the displayed fit line is more defensible: \(27\) or \(41\)? Explain.
Figure for problem 550051

Hints

- Estimate the line’s \(y\)-coordinate at the requested \(x\)-value. - Choose the option consistent with the model rather than an individual extreme point.

Solution

1. Predictions from an informal fit should be near the line’s value. 2. The line’s value is about \(28\). 3. \(27\) is close to \(28\), while \(41\) is not.

Answer

\(27\) is more defensible.
Informal line of best fit 12
In a plot of study time in hours and score in points, a data point is \(6\) vertical units above the fit line. What does that vertical distance mean in context?

Hints

- Identify the variable on the vertical axis. - Translate “above the line” into observed versus predicted language.

Solution

1. Vertical distance is measured in the \(y\)-variable’s units, which are score points here. 2. The observed score is \(6\) points higher than the fitted prediction for that study time.

Answer

The student’s score is \(6\) points above the model’s prediction.
Informal line of best fit 13
Imani calls the displayed blue path \(z\) a “perfect line of fit.” Explain why \(z\) is not a line of fit and describe what a reasonable informal line of fit should do instead.
Figure for problem 550058

Hints

- Inspect the blue path’s shape rather than relying on its label. - Ask whether the display is one straight model or a point-by-point tracing. - A fit should summarize the overall tendency rather than reproduce every local change.

Solution

1. The displayed path connects the observations one by one and changes direction several times. 2. An informal linear fit should be one straight line that summarizes the overall tendency of the cloud rather than tracing every observation. 3. Therefore, \(z\) is not a line of fit; a reasonable fit would be a single increasing straight line through the center of the points.

Answer

Path \(z\) traces the individual observations instead of summarizing them with one straight line. A reasonable fit would be a single increasing line through the center of the cloud.
Informal line of best fit 14
Each panel shows the same scatter plot with a different candidate line of fit. Which line is the most reasonable: a), b), or c)? Explain.
Figure for problem 550145

Hints

- First match the direction and approximate steepness of the point cloud. - A reasonable fit should pass through the middle of the data rather than above or below most points. - Check whether the line represents both the left and right sides of the plot.

Solution

1. The point cloud has a positive, approximately linear pattern. 2. In panel a), the line follows the direction of the cloud, passes through its center, and leaves points on both sides. 3. In panel b), the line is too high for most of the data. In panel c), the line is too flat and misses the pattern at both ends. 4. Therefore, line a) is the most reasonable informal line of fit.

Answer

Line a), because it follows the positive trend and passes through the center of the point cloud with points on both sides.
Informal line of best fit 15
A line of fit for practice sessions \(x\) and predicted performance score \(y\) is \(y=4.5x+62\). According to the model, how many practice sessions correspond to a predicted score of \(80\)? Interpret the result in context.

Hints

- In this problem, the predicted score is known and the input is unknown. - Replace \(y\) with the target score before solving for \(x\). - Interpret the solved x-value using the input variable’s units.

Solution

1. Set the model output equal to the target score: \(80=4.5x+62\). 2. Subtract \(62\): \(18=4.5x\). 3. Divide by \(4.5\): \(x=4\). 4. The model associates about \(4\) practice sessions with a predicted score of \(80\).

Answer

\(4\) practice sessions.
Informal line of best fit 16
At \(x=2\) and \(x=4\), find the difference between the observed \(y\)-value and the value predicted by the blue line. At which \(x\)-value does the line predict too much, and at which does it predict too little?
Figure for problem 560106

Hints

- For each requested \(x\)-value, read the dot's height and the line's height. - Find how far apart the heights are, then decide which is larger.

Solution

1. At \(x=2\), the observation is \(8\) and the line predicts \(7\), a difference of \(1\). The line predicts too little. 2. At \(x=4\), the observation is \(4\) and the line predicts \(5\), also a difference of \(1\). The line predicts too much.

Answer

The difference is \(1\) at each \(x\)-value. The line predicts too little at \(x=2\) and too much at \(x=4\).
Informal line of best fit 17
At \(x=4\), two observations were recorded. What were their \(y\)-values, and what does the line predict? Explain why the line's prediction can differ from both observations.
Figure for problem 560108

Hints

- Look for every dot with the stated \(x\)-coordinate. - Compare both observed heights with the blue line at the same \(x\)-value.

Solution

1. At \(x=4\), the two observations are \(y=4\) and \(y=8\). 2. The fitted line predicts \(y=6\), between those two values. 3. The line gives a typical prediction for that \(x\)-value, not the exact result of every observation.

Answer

Observed: \(4\) and \(8\). Predicted: \(6\). The line summarizes the pattern and need not pass through either observed point.
Informal line of best fit 18
A student wants to move the blue line upward so it passes through the highest dot. Would that make it a better fit for all the points? Explain.
Figure for problem 560109

Hints

- Compare the blue line with the whole cloud, not just the highest dot. - Think about where the line would be relative to the other dots after the move.

Solution

1. Most dots sit close to the blue line, but the dot at \((4, 9)\) is much higher. 2. Moving the entire line up to that dot would put it above most of the other points. 3. It is better to keep the line near the main group rather than force it through the highest dot.

Answer

No. Moving the line up to the highest dot would make it fit most other points worse.
Informal line of best fit 19
Between the smallest and largest observed \(x\)-values, which candidate line, a) or b), shows a change in \(y\) closer to the observed change? Explain using the graph.
Figure for problem 560110

Hints

- Read each line's height at the two stated \(x\)-values. - Compare the amount each line falls with the change in the observations.

Solution

1. The observations decrease by \(9-1=8\) from \(x=1\) to \(x=6\). 2. Line a) drops from \(8\) to \(2\), a decrease of \(6\). 3. Line b) drops from \(7\) to \(5\), a decrease of \(2\). Line a)'s change is closer to the observed decrease.

Answer

a). Its drop of \(6\) units is closer to the observed drop of \(8\) than line b)'s drop of \(2\).
Informal line of best fit 20
In each plot, find the greatest vertical distance between any dot and the blue line. Which line has the smaller greatest distance?
Figure for problem 560111

Hints

- Look for the dot farthest above or below each line, measured vertically. - Check its height against the line at the same \(x\)-value.

Solution

1. In plot a), the largest miss occurs at \(x=5\): the point is at \(y=6\), but the line is at \(y=9\), a vertical distance of \(3\). 2. In plot b), each dot is \(1\) unit from the line, so its greatest vertical distance is \(1\). 3. Line b) has the smaller greatest distance.

Answer

a): \(3\) units. b): \(1\) unit. Line b) has the smaller greatest vertical distance.
Informal line of best fit 21
The table lists measurements from a scatter plot. Which line is a better fit, \(y\approx 2x+1\) or \(y\approx x+3\)? Explain by checking values at the beginning and end of the table. <table><tr><th>\(x\)</th><th>\(1\)</th><th>\(2\)</th><th>\(3\)</th><th>\(4\)</th><th>\(5\)</th><th>\(6\)</th></tr><tr><th>\(y\)</th><td>\(3\)</td><td>\(6\)</td><td>\(6\)</td><td>\(10\)</td><td>\(10\)</td><td>\(14\)</td></tr></table>

Hints

- Substitute an early x-value into each line and compare with the table. - Repeat with a large x-value; see whether either line misses by much more.

Solution

1. The line \(y\approx 2x+1\) predicts \(3\) at \(x=1\) and \(13\) at \(x=6\), close to the observed values \(3\) and \(14\). 2. The line \(y\approx x+3\) predicts \(4\) and \(9\) at those x-values, missing the high-end observation by \(5\). 3. The first line stays closer across the table and is the better fit.

Answer

\(y\approx 2x+1\), because it follows the observations at both ends more closely.
Informal line of best fit 22
Estimate \(y\) when \(x=5\) using the overall pattern of the points. Explain how you estimated it.
Figure for problem 560113

Hints

- Try to picture a straight line that passes through the middle of the full group of points. - Where would that line be when \(x=5\)?

Solution

1. The dots generally rise by about \(2\) for each increase of \(1\) in \(x\), despite a low dot at \(x=4\). 2. A line through the center of the whole cloud is near \(y=12\) at \(x=5\). 3. Using only the two closest dots would give a lower estimate and ignore the larger trend.

Answer

About \(12\). A reasonable estimate close to \(12\) is acceptable if supported by a line through the middle of the points.
Informal line of best fit 23
Compare the predictions at \(x=2\) and \(x=6\). At which \(x\)-value do the observed values differ more from the fitted line's prediction? Use the dots to explain.
Figure for problem 560129

Hints

- Look for both observations at each stated \(x\)-value. - Compare their \(y\)-values with the blue line's height at the same \(x\)-value.

Solution

1. At \(x=2\), the line predicts \(y=4\); the observations are \(2\) and \(6\), each \(2\) units away. 2. At \(x=6\), the line predicts \(y=8\); the observations are \(2\) and \(14\), each \(6\) units away. 3. The observations differ more from the fitted prediction at \(x=6\).

Answer

At \(x=6\). The observations are \(6\) units from the line, compared with \(2\) units at \(x=2\).
Informal line of best fit 24
Use the displayed line to predict \(y\) when \(x=7\). Compare the prediction with the plotted observation at \(x=7\): how far above or below the prediction is the observation?
Figure for problem 550026

Hints

- Read from the fitted line rather than choosing the nearest data point. - Compare the observed \(y\)-coordinate with the fitted line’s value at the same \(x\)-coordinate.

Solution

1. The fit line follows \(y=2x+1\). 2. At \(x=7\), the predicted value is \(2\cdot 7+1=15\). 3. The plotted observation is \((7, 16)\). 4. Since \(16-15=1\), the observation is \(1\) unit above the prediction.

Answer

The prediction is \(y=15\). The observed value is \(y=16\), which is \(1\) unit above the prediction.
Informal line of best fit 25
How should the displayed candidate line be adjusted to better fit the scatter plot?
Figure for problem 550028

Hints

- Compare the errors at small and large \(x\)-values. - A consistent vertical error suggests changing position rather than tilt.

Solution

1. Similar vertical misses across the \(x\)-range indicate that the slope is reasonable. 2. The line is consistently too high. 3. Shift the line downward while keeping approximately the same slope.

Answer

Shift the line downward without substantially changing its slope.
Informal line of best fit 26
Is the displayed candidate line too steep or too shallow for the scatter plot? Explain.
Figure for problem 550029

Hints

- Compare the line with the cloud at both ends of the \(x\)-range. - Decide whether the line rises faster or slower than the data trend.

Solution

1. The line starts too high relative to the cloud. 2. It ends too low relative to the cloud. 3. It rises too slowly, so it is too shallow and needs a larger slope.

Answer

The line is too shallow; its slope should be increased.
Informal line of best fit 27
What correction is needed for the displayed candidate line?
Figure for problem 550030

Hints

- Inspect opposite ends of the \(x\)-range. - A line that overtakes the cloud rises too quickly.

Solution

1. The line begins too low and finishes too high. 2. It rises faster than the data trend. 3. Decrease the slope, rotating the line to be less steep.

Answer

Decrease the slope; the line is too steep.
Informal line of best fit 28
The blue line \(f\) is an informal line of fit for the scatter plot. Choose two convenient grid-intersection points on \(f\) that are not observed data points, use them to find an equation for \(f\), and explain briefly why the displayed line is a reasonable fit for the cloud.
Figure for problem 550031

Hints

- Read two convenient grid intersections from the displayed blue line, not from the black data points. - Use those two line points to determine the slope and intercept. - Then compare the line with the whole cloud: consider closeness and whether the points are reasonably balanced around it.

Solution

1. Two convenient points on the blue line are, for example, \((1,4)\) and \((5,12)\). 2. The slope is \(m=\frac{12-4}{5-1}=\frac{8}{4}=2\). 3. Using \((1,4)\) in \(y=2x+b\) gives \(4=2+b\), so \(b=2\). 4. Thus the line is \(y=2x+2\). 5. The observed points stay close to the line and are distributed on both sides of it across the x-range, so it reasonably summarizes the cloud.

Answer

One valid choice is \((1,4)\) and \((5,12)\), giving \(y=2x+2\). The line is reasonable because the data points stay close to it and lie on both sides across the x-range.
Informal line of best fit 29
Maya draws a line through the leftmost and rightmost data points and calls it the only possible line of fit. Give two reasons this method can fail.

Hints

- Ask whether the two chosen points are typical of the data. - Consider what information about the middle points the method ignores.

Solution

1. The endpoint observations may be unusual and may not represent the center of the cloud. 2. A good informal fit considers all points and balances the overall pattern. 3. Informal fitting can produce more than one reasonable line.

Answer

The endpoint points may be unrepresentative, and a line of fit should summarize the whole cloud rather than be determined by only two observations.
Informal line of best fit 30
Which equation is a better informal fit for the displayed data: \(y=x+2\) or \(y=2x+2\)?
Figure for problem 550033

Hints

- Estimate the overall rise and run of the cloud. - Compare both the slope and vertical placement of each candidate.

Solution

1. The cloud rises by about \(8\) units while \(x\) rises by about \(7\) units. 2. This suggests a slope near \(1\), not \(2\). 3. The equation \(y=x+2\) better matches the center and rate of the cloud.

Answer

\(y=x+2\).
Informal line of best fit 31
The point labeled N is a new accurate observation. Predict how redrawing the informal fit may change its slope.
Figure for problem 550036

Hints

- Consider how the line must tilt to move closer to the new point. - The point is influential because it is far from the center in the \(x\)-direction.

Solution

1. The new point has both a large \(x\)-value and a \(y\)-value above the current line. 2. A line adjusted toward that point will rise more over the full \(x\)-range. 3. The redrawn fit will likely have a larger slope.

Answer

The slope will likely increase.
Informal line of best fit 32
Data were collected only for \(x\)-values from \(10\) to \(30\). Which prediction from the line of fit is more defensible: a prediction at \(x=22\) or at \(x=90\)? Explain.

Hints

- Compare each requested \(x\)-value with the range actually observed. - A fitted trend may not continue far beyond the data.

Solution

1. The value \(x=22\) lies inside the observed range. 2. The value \(x=90\) lies far outside the observed range. 3. The prediction at \(x=22\) is more defensible because it is interpolation rather than distant extrapolation.

Answer

The prediction at \(x=22\) is more defensible.
Informal line of best fit 33
Each panel shows a scatter plot with two candidate fit lines. a) Which line is the better fit in panel a), and which parameter differs between the lines? b) Which line is the better fit in panel b), and which parameter differs between the lines?
Figure for problem 550038

Hints

- In each panel, compare how well the lines match the cloud’s center and direction. - Parallel lines have equal slopes, so vertical separation comes from different \(y\)-intercepts. - Lines with the same \(y\)-intercept must be compared by their slopes.

Solution

1. In panel a), lines \(p\) and \(q\) are parallel, so they have equal slopes. Line \(p\) is centered, while line \(q\) is too high. 2. Choose line \(p\); the lines differ in their \(y\)-intercepts. 3. In panel b), lines \(r\) and \(s\) share a \(y\)-intercept. Line \(r\) follows the cloud, while line \(s\) rises too quickly. 4. Choose line \(r\); the lines differ in their slopes.

Answer

a) Line \(p\); the \(y\)-intercepts differ. b) Line \(r\); the slopes differ.
Informal line of best fit 34
The proposed fit line has exactly \(6\) points above it and \(6\) below it. Is the equal count enough to call it a good fit? Use the graph to explain.
Figure for problem 550039

Hints

- Compare distances as well as the number of points on each side. - A line can split the count evenly and still miss the center of the cloud.

Solution

1. Equal counts alone do not measure how far the points lie from the line. 2. The large one-sided vertical misses show poor centrality. 3. The line should be adjusted to better balance the sizes of the misses.

Answer

No. A good informal fit should consider the sizes and pattern of the vertical misses, not only their counts.
Informal line of best fit 35
Ethan proposes a fit with slope \(0.3\) for the displayed scatter plot. Is that slope plausible?
Figure for problem 550042

Hints

- Estimate the overall rise and run of the cloud. - Check the order of magnitude of the proposed rate.

Solution

1. The cloud’s overall rise-to-run ratio is about \(\frac{15}{5}=3\). 2. A slope of \(0.3\) is ten times smaller. 3. It is not plausible for the displayed trend.

Answer

No. A plausible slope is near \(3\), not \(0.3\).
Informal line of best fit 36
Amara says line \(f\) is better because it has more points above and below it. Leo says line \(g\) is better because its equation has smaller numbers. Without assuming either student is right, what evidence should be used to compare two candidate lines of fit?

Hints

- Think about what a line of fit is supposed to summarize about the entire point cloud. - Ask whether counting points on each side captures how far those points are from the line. - Decide whether the numerical size of an equation’s coefficients says anything by itself about fit quality.

Solution

1. Small coefficients do not make a line a better fit. 2. Simply counting points above and below a line is also insufficient because it ignores how far the points are from the line and how the line follows the cloud. 3. A useful comparison considers whether each line follows the cloud’s direction, passes near its center across the x-range, and keeps vertical misses reasonably small.

Answer

Compare how well each line follows the cloud’s direction, stays near its center across the x-range, and keeps the vertical misses reasonably small.
Informal line of best fit 37
For a candidate line of fit, the table shows the vertical miss \(\text{observed}-\text{predicted}\) at several x-values. <table><tr><th>\(x\)</th><th>\(\text{observed}-\text{predicted}\)</th></tr><tr><td>\(1\)</td><td>\(-3\)</td></tr><tr><td>\(2\)</td><td>\(-2\)</td></tr><tr><td>\(3\)</td><td>\(-1\)</td></tr><tr><td>\(7\)</td><td>\(1\)</td></tr><tr><td>\(8\)</td><td>\(2\)</td></tr><tr><td>\(9\)</td><td>\(3\)</td></tr></table> What pattern do the vertical misses show? Is the line a reasonable fit across the whole x-range, and what change to the line is suggested?

Hints

- Interpret what a negative versus positive value of \(\text{observed}-\text{predicted}\) means. - Compare the signs of the misses at the low and high ends of the x-range. - Think about how rotating a line to make it steeper changes its position on the left and right.

Solution

1. At the smaller x-values, the misses are negative, so the line predicts values that are too high. 2. At the larger x-values, the misses are positive, so the line predicts values that are too low. 3. This systematic change from negative to positive shows that the line is not centered in the cloud across the x-range. 4. The line is too shallow; increasing its slope would lower it relative to the data on the left and raise it relative to the data on the right.

Answer

The misses are negative at small \(x\) and positive at large \(x\), so the line is not a reasonable fit across the whole range. Its slope is too small and should be increased.
Informal line of best fit 38
Estimate a reasonable slope for an informal fit to the displayed scatter plot.
Figure for problem 550048

Hints

- Use representative regions of the cloud rather than isolated extreme points. - Estimate the overall rise divided by the overall run.

Solution

1. Representative centers are near \((2, 7)\) on the left and \((10, 23)\) on the right. 2. The rise is \(23-7=16\), and the run is \(10-2=8\). 3. A reasonable slope is \(\frac{16}{8}=2\).

Answer

A reasonable slope is about \(2\).
Informal line of best fit 39
A student suggests one upward-sloping line of fit for all the points. Compare how \(y\) changes from \(x=1\) to \(x=4\) and from \(x=7\) to \(x=10\). What would the student's line get wrong?
Figure for problem 550050

Hints

- Read the \(y\)-values at the start and end of each stated interval. - Compare the signs of the two changes with what an upward-sloping line would predict.

Solution

1. From \(x=1\) to \(x=4\), the plotted \(y\)-values rise from \(2\) to \(5\), a change of \(+3\). 2. From \(x=7\) to \(x=10\), the plotted \(y\)-values fall from \(15\) to \(12\), a change of \(-3\). 3. An upward-sloping line predicts an increase in both intervals. It misses the falling pattern at larger \(x\)-values.

Answer

\(y\) changes by \(+3\) from \(x=1\) to \(4\), but by \(-3\) from \(x=7\) to \(10\). A single upward-sloping line would incorrectly suggest an increase in the second interval.
Informal line of best fit 40
Both candidate fit lines pass through \((5, 12)\). Which line is the better fit, and why does the shared point not settle the choice?
Figure for problem 550052

Hints

- A single shared point gives no information about tilt. - Compare how each line behaves away from the center.

Solution

1. One point does not determine how a line follows the rest of the cloud. 2. The slope must match the overall direction and rate of change. 3. Line \(a\) tracks the cloud across the \(x\)-range, while line \(b\) is much too steep.

Answer

Line \(a\) is the better fit. Passing through one central point is not sufficient; the slope must also match the overall cloud.
Informal line of best fit 41
A fit was based on \(x\)-values from \(2\) to \(8\). New accurate points from \(9\) to \(12\) continue the same straight pattern. How does this affect the range over which predictions are supported?

Hints

- Update the range of \(x\)-values using the new observations. - A former extrapolation can become interpolation when data are added.

Solution

1. The new points provide observed evidence beyond \(x=8\). 2. They continue the same pattern through \(x=12\). 3. Predictions through \(x=12\) are now supported by interpolation within the expanded data range.

Answer

The data-supported range expands to \(2\le x\le 12\).
Informal line of best fit 42
Give an approximate equation for a straight line that fits the plotted points. Explain how you chose its slope and y-intercept.
Figure for problem 554581

Hints

- Look for a straight line that follows the overall upward pattern. - Compare the points above and below your proposed line across the whole x-range. - Estimate the slope and then the y-intercept of that line.

Solution

1. The points rise about \(2\) units in \(y\) for each \(1\)-unit increase in \(x\). 2. A line with slope \(2\) and intercept \(2.5\) leaves points slightly above and below it. 3. One reasonable equation is \(y\approx 2x+2.5\).

Answer

One reasonable equation is \(y\approx 2x+2.5\). Other nearby equations are acceptable if they follow the trend and pass through the center of the cloud.
Informal line of best fit 43
The scatter plot comes from a simulated battery test, where \(x\) is hours since the test began and \(y\) is battery percentage. Choose a reasonable straight line of fit, give one approximate equation for it, then use your model to estimate the battery percentage at \(x=4.5\) hours.
Figure for problem 554582

Hints

- Choose a line that follows the overall downward direction and stays near the center of the cloud. - Use two readable locations on your proposed line to estimate its rate of change. - Substitute \(4.5\) only after you have a reasonable model.

Solution

1. The cloud has a strong negative linear trend, so a reasonable fit should slope downward through its center. 2. One centered model is \(y\approx -6x+48\). 3. At \(x=4.5\), this model gives \(y\approx -6(4.5)+48=21\). 4. Because the line is an informal fit to scattered data, the prediction is approximate.

Answer

One reasonable model is \(y\approx -6x+48\), which predicts about \(21\%\) battery at \(4.5\) hours. Nearby estimates from other reasonable fit lines are acceptable.
Informal line of best fit 44
Choose a reasonable straight line of fit for the scatter plot. Give two convenient grid-intersection points on your proposed line, write an approximate equation for it, and explain whether a good informal fit must pass through any observed point.
Figure for problem 554583

Hints

- Choose a proposed line that runs through the center of the whole cloud. - Report two readable grid intersections from that proposed line so its slope and intercept can be checked. - Compare the proposed line with the black observations before deciding whether it needs to pass through any one of them.

Solution

1. The cloud is roughly linear and positive, centered near the line \(y=x+5\). 2. Two convenient points on that proposed line are \((1,6)\) and \((8,13)\). 3. Their slope is \(\frac{13-6}{8-1}=1\), giving the equation \(y=x+5\). 4. The observed points lie close to this line and alternate above and below it, even though neither chosen line point has to be an observation. 5. A good informal fit summarizes the cloud and does not need to pass through any observed point.

Answer

One reasonable choice is \((1,6)\) and \((8,13)\), giving \(y\approx x+5\). A good informal fit does not need to pass through any observed point.
Informal line of best fit 45
A fitted model relates elevation \(h\), in thousands of feet, to predicted air temperature \(T\), in degrees Fahrenheit: \(T=78-4h\). The data used to create the fit covered elevations from \(1\) to \(8\) thousand feet. At what elevation does the model predict \(58\,\text{°F}\)? Is using the model for this prediction interpolation or extrapolation?

Hints

- Treat the target temperature as the known output and solve for the input variable. - Keep track of the fact that \(h\) is measured in thousands of feet. - Compare the solved input with the range of elevations used to create the fit.

Solution

1. Set the predicted temperature equal to \(58\): \(58=78-4h\). 2. Subtract \(78\): \(-20=-4h\). 3. Divide by \(-4\): \(h=5\). 4. The model predicts \(58\,\text{°F}\) at an elevation of \(5\) thousand feet. 5. Since \(5\) is within the observed range from \(1\) to \(8\), this is interpolation.

Answer

The model predicts \(58\,\text{°F}\) at \(5\) thousand feet, and this is interpolation.
Informal line of best fit 46
Which displayed line, \(a\) or \(b\), already has a slope that better matches the data cloud? How should that line be adjusted to make it a more reasonable fit?
Figure for problem 550041

Hints

- Separate slope errors from vertical-position errors. - Compare each candidate’s direction with the overall direction of the data cloud.

Solution

1. Line \(a\) already matches the cloud’s rate of change. 2. Its main problem is a small vertical placement error. 3. Shift line \(a\) upward while keeping its slope; line \(b\) would require a larger slope correction.

Answer

Line \(a\); shift it upward while keeping its slope.
Informal line of best fit 47
What does the displayed pattern suggest about the suitability of the straight fit line?
Figure for problem 550044

Hints

- Examine whether the misses are randomly mixed or arranged by \(x\)-region. - A repeated directional pattern in the misses often signals a shape problem.

Solution

1. The line misses the data in an above-below-above pattern. 2. A simple shift or rotation would not remove all three regions of error. 3. The data likely have a curved relationship, so one straight line is unsuitable.

Answer

The systematic pattern suggests nonlinearity rather than a well-fitting straight line.
Informal line of best fit 48
An accurate point lies far from the rest of the data in the \(x\)-direction. Marcus deletes it automatically before fitting. Keiko keeps it automatically. Explain the better decision process.

Hints

- Distinguish an error from a genuine extreme observation. - Consider both mathematical influence and the reason the data were collected.

Solution

1. First verify that the point is accurate and belongs to the same population. 2. Compare fits with and without the point because its extreme \(x\)-value can strongly affect the line. 3. Retain or separate it based on context, not solely because it is visually extreme.

Answer

Check the point’s validity and context, then examine how much it changes the fit; do not automatically keep or delete it.
Informal line of best fit 49
For which interval in the displayed scatter plot is a straight informal fit most useful?
Figure for problem 550056

Hints

- Look for the interval where the rate of change is roughly steady. - A model can be useful locally even when it fails over the full range.

Solution

1. The first interval has an approximately linear pattern. 2. The second interval changes form by leveling off. 3. A straight fit is most useful on \(0\le x\le 6\), not across the full range.

Answer

The interval \(0\le x\le 6\).

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

Building free math worksheets