Exploring Equations of Lines and Correlation Through Scatterplots

Question 1:

First equation – 8y + 90 = 45x

y = (45/8)x – 90/8

Taking x = 12, y = 56.25

Taking y = 0, x = 90/45 = 2

This equation passes through points (2, 0) and (12, 56.25)

Second equation – 45x + 4y = 180

y = (180/4) – (45/4)x

y = 45 – (45/4)x

Taking x = 0, y = 45

Taking y = 0, x = (180/45) = 4

This equation passes through points (0, 45) and (4, 0)

Question 2:

Given points be A(-12, 4) and B(8, -4)

Slope m will be computed as (y2-y1)/(x2-x1) = (-4-4)/(8+12) = (-8)/20 = -2/5

Using equation y = mx + b with point (-12, 4), we will find the intercept b.

4 = m(-12) + b

4 = -12*(-2/5) + b

4 = 24/5 + b

b = -4/5

equation of the line will be y = -2/5x -4/5 slope = -2/5, y-intercept = -4/5 with point (0, -4/5).

Question 3:

Scatterplot A: correlation = 0

Scatterplot B: correlation = -0.99

Scatterplot C: correlation = -0.4

Scatterplot D: correlation = 0.4

Scatterplot E: correlation = -0.7

Scatterplot F: correlation = 0.9

Question 4:

CORREL(Range of influenza cases, range of life expectancy) Correlation coefficient is computed as 0.35

1. Scatterplot does not show any strong linear relationship. Few nations have very high influenza cases like Australia, USA have higher life expectancy whereas the nations with lower number of influenza cases like India, Pakistan often have low life expectancy. Coefficient of correlation is computed as positive at 0.35. There seems to be weak positive linear relationship between life expectancy and number of influenza cases. Nations having advanced healthcare might report both higher cases of influenza with higher life expectancy. So the pattern of data might get impacted by the standards of reporting instead of any biological link.

Assessment Requirements — Brief Summary & Key Pointers

• Q1 (Linear forms & intercepts):

  • Rearrange to slope–intercept form.

  • Find and verify x-/y-intercepts and two points per line.

• Q2 (Line through two points):

  • Compute slope m=y2−y1x2−x1m=\frac{y_2-y_1}{x_2-x_1}m=x2−x1y2−y1.

  • Use y=mx+by=mx+by=mx+b with a given point to find bbb, then write the equation.

• Q3 (Reading correlations from scatterplots):

  • Assign approximate correlation values and justify sign/strength (none, weak, moderate, strong).

• Q4 (Compute & interpret correlation):

  • Use a spreadsheet function (e.g.,

  • Report coefficient, comment on linearity strength, direction, and potential confounders/measurement effects.

How the Academic Mentor Guided the Student 

1. Clarified scope & rubric:
   Matched each question to skills: algebraic manipulation, slope/intercept fluency, visual correlation reading, and quantitative correlation with interpretation.

2. Q1—Form & intercepts:

   • Converted 8y+90=45x8y+90=45x8y+90=45x to y=458x−908y=\frac{45}{8}x-\frac{90}{8}y=845x−890 and checked points (2,0)(2,0)(2,0), (12,56.25)(12,56.25)(12,56.25).

   • Converted 45x+4y=18045x+4y=18045x+4y=180 to y=45−454xy=45-\frac{45}{4}xy=45−445x and verified (0,45)(0,45)(0,45), (4,0)(4,0)(4,0).
     Emphasis: careful arithmetic, explicitly showing intercept logic.

3. Q2—Slope & equation from two points:

   • Computed m=−820=−25m=\frac{-8}{20}=-\frac{2}{5}m=20−8=−52 for A(−12,4),B(8,−4)A(-12,4), B(8,-4)A(−12,4),B(8,−4).

   • Found bbb via 4=(−25)(−12)+b⇒b=−454=(-\tfrac{2}{5})(-12)+b \Rightarrow b=-\tfrac{4}{5}4=(−52)(−12)+b⇒b=−54.

   • Final: y=−25x−45y=-\tfrac{2}{5}x-\tfrac{4}{5}y=−52x−54.
     Emphasis: structure (slope → substitute → solve for bbb → final form).

4. Q3—Interpreting scatterplots:

   • Assigned: A ≈ 0, B ≈ −0.99, C ≈ −0.4, D ≈ 0.4, E ≈ −0.7, F ≈ 0.9.

   • Discussed sign (trend direction) and magnitude (tightness around a line).

5. Q4—Compute and contextualize correlation:

   • Used CORREL(cases, life_expectancy) → r=0.35r=0.35r=0.35.

   • Interpreted as weak positive; noted reporting/health-system effects (e.g., advanced systems report more cases and have higher life expectancy) and cautioned against causal claims.
     Emphasis: statistical literacy—association ≠ causation; data quality matters.

6. Presentation polish:

   • Encouraged clear workings, labeled points, units/contexts, and short justifications beneath results.

   • Suggested a concluding paragraph synthesizing numeric results with real-world interpretation.

Outcome How the Requirements Were Met

• Accuracy: All algebraic rearrangements, intercepts, slope, and line equation were correctly derived and numerically verified.

• Interpretation: Correlations were sensibly matched to visual patterns; r=0.35r=0.35r=0.35 was correctly characterized as weak positive with plausible contextual caveats.

• Communication: Solutions were structured stepwise, showing workings and brief rationale—aligned with assessment expectations.

Learning Objectives Covered

• Convert between linear forms; determine and verify intercepts and points.

• Compute slope and derive a line equation from two points.

• Visually assess correlation sign/strength and relate it to numeric rrr.

• Use spreadsheet functions to compute correlation and interpret results in context.

• Communicate mathematical reasoning clearly and relate findings to real-world data considerations (measurement/reporting bias, non-causality).

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