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Informal line of best fit

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5500478
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
5500258
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.
5500278
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

\(x=6\).
5500348
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.
5500358
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

\((4, 10)\).
5500408
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 \(b\).
5500468
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.
5500518
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.
5500548
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.
5500588
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.
5501458
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.
5545818
The scatter plot shows a roughly linear positive pattern. Choose one reasonable straight line of fit through the center of the point cloud and give an approximate equation for your line in slope-intercept form.
Figure for problem 554581

Hints

- Look for a line that balances the points above and below across the whole x-range. - Estimate the rise compared with the run using two convenient grid locations on your proposed line. - The line summarizes the cloud; it does not need to pass through every observation.

Solution

1. A reasonable fit should follow the upward direction and pass through the middle of the cloud rather than through every point. 2. One line centered in the data has slope about \(2\) and y-intercept about \(3\). 3. One reasonable equation is \(y\approx 2x+3\).

Answer

One reasonable equation is \(y\approx 2x+3\). Other nearby equations are acceptable if they describe a line centered in the cloud with about the same upward slope.
5545838
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.
5500268
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.
5500288
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.
5500298
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.
5500308
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.
5500318
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.
5500328
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.
5500338
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

\(y=x+2\).
5500368
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.
5500378
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.
5500388
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.
5500398
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.
5500428
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 \(0.3\).
5500438
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.
5500458
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.
5500488
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 \(2\).
5500528
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.
5500538
Choose two convenient points from the displayed negative fit line and use them to find its slope.
Figure for problem 550053

Hints

- Use points on the line that fall exactly on readable grid locations. - Compute vertical change over horizontal change in left-to-right order.

Solution

1. Two clear points on the line are \((0, 12)\) and \((6, 3)\). 2. The slope is \(\frac{3-12}{6-0}=\frac{-9}{6}=-1.5\). 3. The fit line has slope \(-1.5\).

Answer

The slope is \(-1.5\).
5500578
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 \(2\le x\le 12\).
5545828
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.
5500418
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.
5500448
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.
5500508
The scatter plot shows operating time and output measurements from two different types of machines. Explain why one straight fit through all the points may be misleading.
Figure for problem 550050

Hints

- Ask whether the combined line passes through regions containing actual data. - Consider whether the groups may follow different relationships.

Solution

1. One combined line can fall between the clusters where few or no observations occur. 2. It can hide differences between the machine types. 3. Separate fits or separate descriptions may better represent the two groups.

Answer

One combined line may describe neither machine type well; the clusters should be examined separately.
5500558
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.
5500568
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 \(0\le x\le 6\).

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