Scatter plots and two-variable data

1. Hourly pay = pay ÷ hours. Use the first column: \frac{\$86}{5\text{ hours}} = \$17.20 \text{ per hour}
2. Check that the rate is the same in another column: \frac{\$172}{10} = \$17.20 \qquad \frac{\$240.80}{14} = \$17.20
3. Or compare two columns: from 5 to 8 hours (3 more hours) the pay goes up 137.60 - 86 = 51.60 dollars, and 51.60 \div 3 = 17.20.

Marcus earns $17.20 per hour.

Check: 17.20 \times 8 = 137.60 ✓, matching the table.

Tip: In a table of two variables, a constant rate means \dfrac{\text{change in } y}{\text{change in } x} is the same for every pair of columns. Be sure to divide the change in pay by the change in hours.

1. Find week 2 on the horizontal axis and go straight up to the point. Read across to the vertical axis: the level is 59\text{ ft}.
2. Find week 6 the same way: the level is 47\text{ ft}.
3. The drop is the difference between the two readings: 59 - 47 = 12

The water level dropped 12 feet.

Check: Add the weekly changes from week 2 to week 6: (59-54)+(54-52)+(52-50)+(50-47) = 5+2+2+3 = 12 ✓

Tip: "How much did it change" always means subtract the two readings. Read both values from the scale; never answer with a single reading, and make sure you start at the time the question names.

1. Read the plot from left to right (more TV). The points start high on the left (about 9 hours of sleep) and fall to the lower right (about 5.5 hours of sleep).
2. When one variable tends to decrease as the other increases, the association is negative.
3. Is it perfect? No. The points lie close to a line, but not exactly on one. For example, at 1 hour of TV there are two different sleep values (8.5 and 9).
4. So the best description is: a negative association: people who watch more TV tend to sleep less.

Check: Compare two ends: at 0.5 hour of TV, sleep is 9 hours; at 5.5 hours of TV, sleep is 5.5 hours. Up on the x-axis, down on the y-axis ✓

Tip: Positive and negative describe the direction of the trend, not whether it is good or bad. Also, a negative association does not prove that TV causes less sleep; it only shows that the two tend to move in opposite directions.

1. The trend line starts at (0, 24) and ends at (10, 4). It falls 24 - 4 = 20 thousand dollars over 10 years, so it falls 20 \div 10 = 2 thousand dollars per year.
2. The equation of the line is y = 24 - 2x, where x = age in years and y = value in thousands of dollars.
3. Substitute x = 4: y = 24 - 2(4) = 24 - 8 = 16
4. The units are thousands of dollars: 16 \times 1{,}000 = \$16{,}000.

The predicted value is $16,000.

Check: Reading the graph: at age 4 the dashed line passes through the gridline at 16 ✓ (The actual 4-year-old car in the data is worth $17,000 — a little above the line, which is normal.)

Tip: A prediction comes from the trend line, not from the nearest data point. Make sure you use the line when the question says "based on the trend line."

1. Read the temperatures: 6 am 48, 8 am 53, 10 am 62, 12 pm 68, 2 pm 72, 4 pm 70, 6 pm 61.
2. Find the change for each 2-hour period:

3. Only the last two periods are decreases. The largest decrease is -9, between 4 pm and 6 pm.

Check: On the graph, the segment from 4 pm to 6 pm is the steepest one going down ✓

Tip: Read the direction as well as the size: +9 and -9 are the same size, but one is a rise and the other is a drop. Steepness shows how fast something changes; the direction of the segment shows whether it goes up or down.

1. Every 2 weeks the height rises by 7 - 4 = 3 inches.
2. Rate per week: \frac{3\text{ in}}{2\text{ weeks}} = 1.5 \text{ inches per week}
3. Starting height (week 0) is 4 inches. The equation is h = 4 + 1.5w.
4. At week 10: h = 4 + 1.5(10) = 4 + 15 = 19

The plant will be 19 inches tall.

Check: Extend the table by 2s: week 8 → 13 + 3 = 16, week 10 → 16 + 3 = 19 ✓

Tip: When the x-values in a table go up by more than 1, divide the change in y by the change in x to get the rate per unit. Then add the starting value.

1. The question asks for a prediction from the trend line, not from a single data point. (No electrician in the data has exactly 8 years.)
2. Find where the line crosses gridlines. It starts at (0, 18) and ends at (12, 42). Over 12 years it rises 42 - 18 = 24 dollars, so it rises \frac{24}{12} = 2 \text{ dollars per year}
3. Start at $18 and add $2 for each of 8 years: 18 + 2(8) = 18 + 16 = 34
4. On the graph: go up from 8 to the dashed line, then left to the axis. You land at $34.

The predicted wage is $34 per hour.

Check: On the line, 7 years gives 18 + 2(7) = 32 and 9 years gives 18 + 2(9) = 36. Halfway between $32 and $36 is $34 ✓

Tip: When a question says "based on the trend line," read the line, not the nearest dot. Finding the line's rate (rise over run between two clear gridline points) lets you predict any value exactly.

1. The increase per year is the slope of the trend line: m = \frac{\text{change in wage}}{\text{change in years}} (The slope formula is on the GED formula sheet.)
2. Choose two clear points on the line: (0, 18) and (12, 42).
3. Compute: m = \frac{42 - 18}{12 - 0} = \frac{24}{12} = 2
4. The units are dollars per hour per year of experience.

The wage increases by about $2 per hour for each year of experience.

Check: Use other points on the line: at 6 years the line is at $30, and \frac{30 - 18}{6 - 0} = 2 ✓

Tip: "Increase for each" or "per" signals the slope. The y-intercept ($18) is a starting value, and dividing one y-value by one x-value (42 \div 12) is not a slope unless the line goes through the origin.

1. List the change for each pair of consecutive weeks (later level minus earlier level):

2. The question asks about a decrease, so look only at the negative changes.
3. The largest drop is -5 feet, from week 2 (59 ft) to week 3 (54 ft).

The level decreased the most between week 2 and week 3.

Check: On the graph, the segment from week 2 to week 3 is the steepest downhill segment ✓ The segment from week 7 to week 8 is also steep, but it goes up.

Tip: On a line graph, the steepest segment shows the fastest change. Then check its direction: uphill is an increase, downhill is a decrease. Don't confuse the biggest change with the lowest point.

1. The y-intercept is the point on the line where x = 0.
2. Translate using the axis labels: x is hours of use and y is battery remaining (%).
3. So (0, 100) says: after 0 hours of use, the model predicts 100% battery remaining.
4. In plain words: the trend line predicts that a phone starts fully charged before it is used.

The point means a phone that has been used for 0 hours is predicted to have 100% battery.

Check: That makes sense in the real world: the reviewer charged the phones fully before the test, and the line falls from there ✓

Tip: To interpret any point (a, b) on a graph, say it as a sentence: "When [x-label] is a, [y-label] is b." The y-intercept is the starting value; the slope (-8) is the rate of change. Keep those two roles separate.

1. Describe the pattern: from 500 to 1,100 square feet, rent rises steadily from about $850 to about $1,450. Larger apartments tend to cost more: a positive trend.
2. Use the trend to predict rent for each size. A 1,200-square-foot apartment should rent for about $1,500 (between $1,450 at 1,100 sq ft and $1,600 at 1,300 sq ft).
3. But the point at 1,200 sq ft shows a rent of only $900, about $600 below the trend. It is far from the pattern.

The outlier is the 1,200-square-foot apartment that rents for $900.

Check: Every other point is within about $50 of the steady upward pattern; only (1,200, 900) breaks it ✓

Tip: An outlier is about position relative to the pattern. The cheapest apartment (500 sq ft, $850) and the most expensive (1,300 sq ft, $1,600) are extremes, but both fit the trend. In real life, an outlier might signal a special case (here, maybe an apartment that needs repairs).

1. The total includes a fixed fee, so dividing a total by its hours will not give the hourly rate.
2. Instead, find how much the charge increases per extra hour: \frac{190 - 125}{2 - 1} = \frac{65}{1} = 65
3. Test another pair: from 3 to 5 hours, \frac{385 - 255}{5 - 3} = \frac{130}{2} = 65 The rate of change is the same, so it is the hourly rate.
4. The fixed fee is what is left at 1 hour: 125 - 65 = 60 dollars.

The hourly rate is $65 per hour.

Check: Fee + rate \times hours: 60 + 65(5) = 60 + 325 = 385 ✓, matching the table.

Tip: When a table has a starting fee or starting amount, the rate is the change in y divided by the change in x, not y \div x. If y \div x gives different results for different columns (125, 95, 85, 77), the relationship is not proportional.

1. y-intercept: the trend line crosses the vertical axis at (0, 24), so b = 24.
2. Slope: use two points on the line, (0, 24) and (10, 4): m = \frac{4 - 24}{10 - 0} = \frac{-20}{10} = -2
3. Put them into slope-intercept form y = mx + b: y = -2x + 24

The equation is y = -2x + 24.

Check: Substitute x = 10: y = -2(10) + 24 = 4 ✓, which matches where the line ends.

Tip: A line that goes down from left to right must have a negative slope — eliminate any choice with a positive slope right away. Both the slope formula and y = mx + b are on the GED formula sheet.

1. The trend line has the form y = mx + b (slope-intercept form, on the formula sheet).
2. y-intercept b: the line crosses the vertical axis at (0, 100), so b = 100.
3. Slope m: use two points where the line crosses gridlines, (5, 60) and (10, 20): m = \frac{20 - 60}{10 - 5} = \frac{-40}{5} = -8 The slope is negative because the line falls from left to right.
4. Put them together: y = -8x + 100

The equation is y = -8x + 100.

Check: Substitute x = 10: -8(10) + 100 = 20 ✓, matching the point (10, 20) on the line.

Tip: Always check the sign of the slope by looking at the graph: falling line, negative slope. Then test your equation with one point you can read from the line. An equation that fails the test is wrong, no matter how close it looks.

1. The miles jump by different amounts (3, 4, 3), so compare changes: \frac{14.50 - 8.50}{5 - 2} = \frac{6}{3} = 2 \qquad \frac{22.50 - 14.50}{9 - 5} = \frac{8}{4} = 2 Each mile adds $2.
2. Find the starting (base) fare by going back from 2 miles to 0 miles: 8.50 - 2(2) = 8.50 - 4 = 4.50
3. Write the rule: \text{fare} = 4.50 + 2m.
4. For m = 20: 4.50 + 2(20) = 4.50 + 40 = 44.50

The fare is $44.50.

Check: Start from the last column: 12 miles cost $28.50, and 8 more miles add 8 \times \$2 = \$16: 28.50 + 16 = 44.50 ✓

Tip: A fare with a base charge is not proportional, so you cannot just scale ($8.50 for 2 miles does not mean $85 for 20 miles). Find the rate from the changes, then the starting value, then predict.

1. In y = 0.6x + 18, the number multiplied by x is the slope: the change in y for each increase of 1 in x.
2. One more year (x goes up by 1) makes y go up by 0.6.
3. But y is measured in thousands of people, so an increase of 0.6 means 0.6 \times 1{,}000 = 600 \text{ people}
4. So the population grows by about 600 people per year.

The best interpretation is: the town's population increases by about 600 people each year.

Check: Compare 2010 (x = 0, y = 18, so 18,000 people) and 2011 (x = 1, y = 18.6, so 18,600 people): 18{,}600 - 18{,}000 = 600 ✓

Tip: Always attach units to the slope: (units of y) per (unit of x). When a variable is "in thousands" or "in millions," convert at the end, or your answer will be off by a factor of 1,000.

1. A positive association means fires with more firefighters also tend to have more damage.
2. An association alone does not show that one variable causes the other.
3. Think about a third factor: the size of the fire. A bigger fire causes more damage, and a bigger fire also makes the city send more firefighters. The size of the fire drives both numbers up.
4. So the best-supported conclusion is: fires with more firefighters tend to have more damage, probably because larger fires need more firefighters and cause more damage; the data do not show that firefighters cause damage.

Answer: larger fires likely explain both; the association does not prove firefighters cause damage.

Check: The correct choice describes what tends to happen and names a third factor (fire size) that raises both numbers. It never claims that one variable causes the other ✓

Tip: On the GED, "best supported" conclusions about scatter plots use words like "tend to" or "are associated with." Choices that say "causes," "proves," or "always" usually go too far.

1. The trend line passes through (0, 24) and (10, 4). Slope: m = \frac{4 - 24}{10 - 0} = -2 so the equation is y = -2x + 24 (y in thousands of dollars).
2. $14,000 is y = 14. Substitute: 14 = -2x + 24
3. Subtract 24 from both sides: -10 = -2x
4. Divide both sides by -2: x = 5

The car will be worth $14,000 at about 5 years old.

Check: -2(5) + 24 = -10 + 24 = 14 ✓ On the graph, the line crosses the 14 gridline right above 5 years ✓

Tip: This is a "work backwards" question: you know y and need x. Substitute and solve, or read the graph across from the y-value to the line and then down. Always check that the answer makes sense on the graph.

1. Pick two clear points on the line. It crosses gridline intersections at (20, 90) and (50, 30).
2. Slope: m = \frac{30 - 90}{50 - 20} = \frac{-60}{30} = -2
3. y-intercept: the graph's horizontal axis starts at 10, so the line never touches x = 0 on the picture. Use y = mx + b with the point (20, 90): 90 = -2(20) + b \Rightarrow 90 = -40 + b \Rightarrow b = 130
4. The equation is y = -2x + 130

The trend line is y = -2x + 130.

Check: Test the other point (50, 30): -2(50) + 130 = -100 + 130 = 30 ✓

Tip: When the axis does not start at 0, the value where the line meets the left edge is not the y-intercept. Find the slope first, then plug one point into y = mx + b to solve for b.

1. The 50 is a number of cups, so it is a y-value. We need the matching x (temperature).
2. Set y = 50: 50 = -2x + 130
3. Subtract 130 from both sides: -80 = -2x
4. Divide both sides by -2: x = 40

The stand would sell about 50 cups at 40°F.

Check: -2(40) + 130 = -80 + 130 = 50 ✓ On the graph, the point (40, 52) is right next to the line near 50 cups.

Tip: Before plugging in, label the given number: is it an input (x) or an output (y)? Putting 50 in for x gives -2(50) + 130 = 30, which answers a different question: how many cups at 50°F.