Scatter Plots and Two-Way Tables: practice solutions, Grade 8 – download the PDF

1 Plot and describe ★★★

1. Temperature, because it is the variable that may influence sales (independent variable). Cups sold go on the vertical axis.
2. \((60, 12)\) and \((85, 38)\).
3. As temperature rises, the number of cups sold rises, so the association is positive.

2 Name that association ★★★

1. Negative: older cars are usually worth less.
2. No association: shoe size has no connection with quiz results.
3. Positive: more practice tends to mean more baskets.
4. Positive: a longer distance takes more time.
5. Negative: more absences tend to go with lower grades.

3 Reading a scatter plot ★★★

1. Negative association: older bikes tend to cost less.
2. The point at \(x = 4\) is \((4, 6)\), so the bike costs 600 dollars.
3. Points with \(y > 5\) are \((1, 9)\), \((2, 8)\), \((3, 8)\), \((4, 6)\): 4 bikes.

4 Complete the totals ★★★

Row totals: \(14 + 9 = 23\) and \(6 + 11 = 17\). Column totals: \(14 + 6 = 20\) and \(9 + 11 = 20\). Grand total: \(23 + 17 = 40\), which matches \(20 + 20 = 40\).

5 Spot the outlier ★★★

The pages increase by about 12 each day: 12, 25, 36, 48, then 73, 85. The pair \((5, 11)\) is far below the expected value of about 60 for day 5, so it is the outlier. It could be a recording mistake.

6 Slope of a trend line ★★★

1. \(m = \dfrac{35 - 15}{5 - 1} = \dfrac{20}{4} = 5\).
2. \(15 = 5(1) + b\), so \(b = 10\).
3. \(y = 5x + 10\). Check with \((5, 35)\): \(5(5) + 10 = 35\). Correct.

7 Use the equation ★★★

1. \(y = 3(10) + 8 = 38\) songs.
2. \(3x + 8 = 50\), so \(3x = 42\) and \(x = 14\). After 14 weeks.

8 Build the scatter plot ★★★

1. The points fall from left to right: negative, linear association (see the graph).
2. \(m = \dfrac{23 - 38}{9 - 6} = -5\). Then \(38 = -5(6) + b\), so \(b = 68\). The line is \(y = -5x + 68\).
3. \(y = -5(8.5) + 68 = 25.5\). About 25.5 minutes.

9 Which line fits best? ★★★

1. Line A, \(y = 1.6x + 0.5\). It follows the direction of the points, with points on both sides.
2. Line B, \(y = x + 4\), is too flat and lies on or above every point (it touches only the last one), so it overestimates. Line C, \(y = 2x - 3\), lies below every point, so it underestimates.

10 Interpolate or extrapolate? ★★★

1. \(4(7) + 12 = 40\) cm.
2. \(4(30) + 12 = 132\) cm.
3. The week 7 prediction (interpolation), because 7 lies inside the measured weeks 1 to 10. Week 30 is extrapolation: the seedling will not keep growing at the same pace forever.

11 Scooter rentals ★★★

1. Each extra minute adds about 0.40 dollars (40 cents) to the cost.
2. When \(x = 0\), the cost is 3 dollars: a starting fee for unlocking the scooter.
3. \(0.40x + 3 = 10\) gives \(0.40x = 7\), so \(x = 17.5\). You can ride up to 17.5 minutes.

12 Honor roll and band ★★★

1. Row totals: 30 and 50. Column totals: 40 and 40.
2. \(\dfrac{18}{30} = 60\%\).
3. \(\dfrac{22}{50} = 44\%\).
4. 60% and 44% are noticeably different, so there is an association: band students are more likely to be on the honor roll in this sample.

13 Fill in from clues ★★★

Bus and pet: 14. Bus and no pet: \(35 - 14 = 21\). Walk and pet: \(24 - 14 = 10\). Walk and no pet: \(25 - 10 = 15\). No pet total: \(21 + 15 = 36 = 60 - 24\).

14 Cats or dogs? ★★★

Grade 7: \(\dfrac{24}{60} = 40\%\) like cats. Grade 8: \(\dfrac{20}{50} = 40\%\) like cats. The percents are equal (and 60% for dogs in both), so there is no association between grade and favorite pet.

15 The outlier effect ★★★

1. \((7, 5)\).
2. \(4(7) + 8 = 36\) passes.
3. \(36 - 5 = 31\) passes below the prediction.
4. For example, the pool was closed that week because of a storm. An outlier often has a special cause.

16 Two clusters ★★★

1. Cluster 1: 5 players practicing 1 to 3 hours, scoring 40% to 52%. Cluster 2: 5 players practicing 7 to 9 hours, scoring 70% to 85%. There is a gap between 3 and 7 hours with no data.
2. It would be unreliable: no player practices around 5 hours, so the prediction would not be based on data there. The data probably come from two groups (casual players and team players).

17 Resale value ★★★

1. \(m = \dfrac{400 - 640}{4 - 1} = -80\). Then \(640 = -80(1) + b\), so \(b = 720\). \(y = -80x + 720\).
2. The phone loses about 80 dollars of value each year. The intercept 720 dollars is the predicted value at purchase (when \(x = 0\)).
3. \(-80x + 720 = 0\) gives \(x = 9\). Nine years is far beyond the data (extrapolation), and real phones often become obsolete sooner, so it is only a rough estimate.

18 Fit your own line ★★★

1. Positive and linear.
2. Sample answer: use \((10, 100)\) and \((60, 600)\). \(m = \dfrac{500}{50} = 10\) and \(b = 100 - 10(10) = 0\), so \(y = 10x\). (Nearby lines are also acceptable.)
3. \(y = 10(45) = 450\) calories.
4. About 10 more calories are burned for each extra minute of jogging.

19 Association in a table ★★★

1. \(\dfrac{66}{100} = 66\%\) and \(\dfrac{48}{100} = 48\%\).
2. The difference is 18 percentage points, so there is an association.
3. \(\dfrac{66}{114} \approx 57.9\%\).

20 Relative frequency table ★★★

1. \(0.18 \times 500 = 90\), \(0.42 \times 500 = 210\), \(0.12 \times 500 = 60\), \(0.28 \times 500 = 140\).
2. Bus riders: \(\dfrac{90}{300} = 30\%\). Walkers: \(\dfrac{60}{200} = 30\%\).
3. The percents are equal, so there is no association.

21 Cause or coincidence? ★★★

1. \(1.5x - 60 = 0\) gives \(x = 40\). At 40 °F.
2. Each degree of warming is associated with about 1.5 dollars more in sales.
3. Association is not causation. Warm weather leads to both more ice cream sales and a warmer town; ice cream does not cause the heat.