choosing the model that fits the data: linear, quadratic or exponential

Grade 11 Math Worksheet

Circle the best answer.

1. 1.

   0 → 50; 1 → 40; 3 → 25.6. Tala tracks a vehicle's value in thousands of dollars at years 0, 1 and 3. Which equation fits all three rows?

   • y = 50 − 10x
   • y = 50(0.64)^x
   • y = 50(0.8)^x

2. Read this first

   Table, week → cm

   1 → 2.1; 2 → 2.6; 3 → 3.9; 4 → 6.2; 5 → 9.1; 6 → 12.8; 7 → 17.0. Line: y = 2.51x − 2.37, r = 0.97. Residuals: +2.0, 0.0, −1.3, −1.5, −1.1, +0.1, +1.8.

3. 2.

   Luz fits a line to a plant's height by week. Technology gives r = 0.97, but the residuals form a U. Which conclusion is best?

   • The line fits: the residuals are small, from −1.5 to +2.0
   • The line fits well: r = 0.97 is close to 1, so the residuals do not matter
   • The line is a poor model: a high r can hide a curve, so a quadratic is better
4. 3.

   Kira fits a line and an exponential to the same six points. Both have r above 0.96. Which model should Kira use?

   • Either one: both have r above 0.96, so they are equally good for these points
   • The exponential: its residuals are small and scattered; the line's form a U
   • The line: it has fewer parts, and its residuals are about as small as the curve's
5. 4.

   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
   • The line is centered, but the errors grow with x, so large-x predictions are less reliable
   • A curve fits better, because the residuals alternate above and below 0 as x grows
   • The line fits equally well across the x-values, because the residuals average about 0 across the eight points

6. 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.

7. 5.

   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?

   • Delete the point at once, because removing it raises r from 0.43 to 0.99
   • Check whether (9, 3) is a mistake or a real case, then compare fits with and without it
   • Keep the whole data set and use the line, because r is simply what the data give, however odd the point looks

8. 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.

9. 6.

   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 power model: ln y against x is a line, so y is a power of x
   • A line: r = 0.935 is already close to 1, so no change to the model is needed
10. 7.

    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?

    • 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
    • Too high: the line goes through the middle of the data, so it overestimates
11. 8.

    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 = 0.34x + 15, because only the slope changes
    • y = 0.34x + 2.1, because both numbers are divided by 7
    • y = 16.8x + 15, because there are 7 days in a week
12. 9.

    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 starts higher, because 1.09 is bigger than 1.06
    • Both start at 2,400; the second grows 9% a year, the first 6%
    • The second adds 3% more cardholders each year, a constant amount
13. 10.

    Technology fits y = 800(0.75)^(x/2) to a phone's resale value in dollars after x years. Which description does Luz choose?

    • Starts at 800 and falls about 13.4% each year, 25% every 2 years
    • Starts at 800 and falls 25% each year, so about 600 are left after 1 year
    • Starts at 800 and falls 12.5% each year, half of the 25% over 2 years
    • Starts at 0.75 and is multiplied by 800 every 2 years