choosing the model that fits the data: linear, quadratic or exponential
Grade 11 Math Worksheet
Circle the best answer.
2 → 48; 4 → 80; 8 → 96; 10 → 80. Bo's lemonade stand tried four prices and recorded the daily revenue. Which model fits, and where does revenue peak?
- Quadratic: revenue peaks at $8, because 96 is the largest value in the table
- Exponential: revenue grows fast at first, then levels off near the $96 peak
- Quadratic: revenue peaks at $10, the highest price tried
- Quadratic: it peaks near $7, halfway between the equal $4 and $10 rows
2 → 18; 5 → 144. Zee knows a lab dish grows by a constant factor each hour, and has two readings. Which equation y = a(b)^x passes through both?
- y = 18(8)^x
- y = 18(2)^x
- y = 4.5(2)^x
1 → 4.3; 2 → 5.7; 3 → 8.4; 4 → 11.2; 5 → 16.4. Ren has a video's weekly views for weeks 1 to 5. Which equation is within 1 of every one of the five values?
- y = 3(1.4)^x
- y = 3x + 1.4
- y = 3(1.5)^x
Over 20 summer days, a town's ice-cream sales and sunburn cases have r = 0.88. Which conclusion does Jun say the data support?
- Selling ice cream causes sunburns, because r = 0.88 is so close to 1 here
- Both rise on hot days, so r shows a link but not a cause
- Sales and sunburns are barely related, because r is not exactly equal to 1
Milo fits a line to eight points. The residuals alternate in sign and the later ones are much larger. Which conclusion is best?
- A curve fits better, because the line's r is just 0.93
- A curve fits better, because the residuals alternate above and below 0 as x grows
- The line is centered, but the errors grow with x, so large-x predictions are less reliable
- The line fits equally well across the x-values, because the residuals average about 0 across the eight points
Read this first
Points
(1, 2.4), (2, 3.1), (3, 4.9), (4, 5.2), (5, 6.8), (6, 7.1), (7, 8.9), (9, 3). With all eight, r = 0.43. Without (9, 3), r = 0.99.
A line through eight points has r = 0.43. Without the point (9, 3), the other seven give r = 0.99. What should Tala do first?
- Check whether (9, 3) is a mistake or a real case, then compare fits with and without it
- Delete the point at once, because removing it raises r from 0.43 to 0.99
- Keep the whole data set and use the line, because r is simply what the data give, however odd the point looks
Table, x → y → ln y
1 → 6 → 1.79; 2 → 10 → 2.30; 3 → 17 → 2.83; 4 → 29 → 3.37; 5 → 50 → 3.91; 6 → 86 → 4.45. r for (x, y) is 0.935; r for (x, ln y) is 0.9999.
Noah fits y against x and then ln y against x. Technology gives the two r values shown. Which model does the data support?
- A quadratic: the points curve upward, so a parabola should fit them well
- An exponential: ln y is almost linear in x, so y grows by a steady factor
- A line: r = 0.935 is already close to 1, so no change to the model is needed
- A power model: ln y against x is a line, so y is a power of x
Pia fits a line to six points and gets the residuals shown, a U. Just beyond x = 6, will the line likely be too high or too low?
- Too high: the line goes through the middle of the data, so it overestimates
- Right on target: the residuals average 0, so the line is exact
- Too low: the data curve upward, so the line falls further below them
Esme's fit y = 2.4x + 15 gives a vine's height in cm after x weeks. Esme rewrites it with x in days. Which equation matches?
- y = 16.8x + 15, because there are 7 days in a week
- y = 0.34x + 2.1, because both numbers are divided by 7
- y = 0.34x + 15, because only the slope changes
Fits y = 2,400(1.06)^x and y = 2,400(1.09)^x give two towns' library cardholders after x years. Which statement does Ari make comparing them?
- The first grows faster, because 1.06 is closer to 1
- The second adds 3% more cardholders each year, a constant amount
- The second starts higher, because 1.09 is bigger than 1.06
- Both start at 2,400; the second grows 9% a year, the first 6%