Mathematics 10 — Exercise Review · Q4 (i–iii)

Full question

The table lists weekly exercise time in hours and resting heart rate in beats per minute. Draw a scatter diagram and a line of best fit, describe the correlation, and estimate Sakeena’s resting heart rate if she exercises for 6.5 hours per week.

Step-by-step solution

1. Each table column forms ONE ordered pair (x,y). Keep each pair together.

2. Label the horizontal and vertical axes, choose uniform scales, then plot every pair.

3. Draw a straight line through the middle of the point cloud, with points on both sides. This is an estimate; different sensible hand-drawn lines give slightly different predictions.

4. Strong negative correlation: y tends to fall as x rises.

5. For reproducible estimates the drawn line uses least squares. Its slope is Σ(x−x̄)(y−ȳ)/Σ(x−x̄)²; intercept is ȳ−slope×x̄.

   $$y=\frac{4849}{55}-\frac{29\times x}{11}$$

6. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{\left(-377\right)\times 1}{1\times 22}$$

7. Calculate this operation before combining the remaining terms.

   $$\left(-377\right)\times \frac{1}{22}=-\frac{377}{22}$$

8. For multiplication of fractions, multiply the numerators together and the denominators together.

   $$\frac{4849\times 1}{1\times 55}$$

9. Calculate these two products separately.

   $$\frac{4849}{55}$$

10. Calculate this operation before combining the remaining terms.

    $$4849\times \frac{1}{55}=\frac{4849}{55}$$

11. Substitute the given horizontal-axis value in the line equation.

    $$-\frac{377}{22}+\frac{4849}{55}$$

12. Calculate this operation before combining the remaining terms.

    $$\left(-29\right)\times 13=-377$$

13. Calculate this operation before combining the remaining terms.

    $$11\times 2=22$$

14. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-29\right)\times 13}{11\times 2}$$

15. Calculate these two products separately.

    $$-\frac{377}{22}$$

16. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-29\right)\times 1}{1\times 11}$$

17. Calculate these two products separately.

    $$-\frac{29}{11}$$

18. Calculate this operation before combining the remaining terms.

    $$\left(-29\right)\times \frac{1}{11}=-\frac{29}{11}$$

19. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{13\times 1}{1\times 2}$$

20. Calculate these two products separately.

    $$\frac{13}{2}$$

21. Calculate this operation before combining the remaining terms.

    $$13\times \frac{1}{2}=\frac{13}{2}$$

22. Calculate this operation before combining the remaining terms.

    $$\left(-\frac{29}{11}\right)\times \frac{13}{2}=-\frac{377}{22}$$

23. To add these fractions, first use a common denominator: the least common multiple of their denominators.

    $$\mathrm{LCD}=110$$

24. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$-\frac{377}{22}=\frac{\left(-377\right)\times 5}{22\times 5}$$

25. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{\left(-1885\right)\times 1}{1\times 110}$$

26. The numerator and denominator have this greatest common divisor. Divide BOTH by it.

    $$\mathrm{GCD}=5$$

27. Show the two divisions explicitly.

    $$\frac{-\frac{1885}{5}}{\frac{110}{5}}$$

28. Evaluate the numerator division.

    $$-\frac{1885}{5}=-377$$

29. Evaluate the denominator division.

    $$\frac{110}{5}=22$$

30. Calculate this operation before combining the remaining terms.

    $$\left(-1885\right)\times \frac{1}{110}=-\frac{377}{22}$$

31. Calculate the converted numerator; the denominator is now common.

    $$-\frac{1885}{110}$$

32. Multiply this numerator AND denominator by the same factor. Its value stays unchanged.

    $$\frac{4849}{55}=\frac{4849\times 2}{55\times 2}$$

33. Calculate the converted numerator; the denominator is now common.

    $$\frac{9698}{110}$$

34. Calculate this operation before combining the remaining terms.

    $$-1885+9698=7813$$

35. For multiplication of fractions, multiply the numerators together and the denominators together.

    $$\frac{7813\times 1}{1\times 110}$$

36. Calculate these two products separately.

    $$\frac{7813}{110}$$

37. Calculate this operation before combining the remaining terms.

    $$7813\times \frac{1}{110}=\frac{7813}{110}$$

38. Keep the common denominator and add the signed numerators.

    $$\frac{-1885+9698}{110}=\frac{7813}{110}$$

39. Calculate this operation before combining the remaining terms.

    $$-\frac{377}{22}+\frac{4849}{55}=\frac{7813}{110}$$

Answer

Strong negative correlation. Estimated Heart rate≈71.03.