Question

the scatter plot shows the time spent studying, x, and the quiz score, y, for each of 23 students. use the scatter plot to answer the parts below. (note that you can use the graphing tools to help you approximate the line.)
scratch area (not part of answer)
(a) write an approximate equation of the line of best fit. round the coefficients to the nearest hundredth.
( y = )
(b) using your equation from part (a), predict the quiz score for a student who spent 70 minutes studying. round your prediction to the nearest hundredth.

Explanation:

Step1: Estimate line of best fit

From the scatter plot, we can estimate two points on the line of best fit. Let's assume two points: (20, 25) and (80, 75).

Step2: Calculate slope (m)

The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substituting the points: $m = \frac{75 - 25}{80 - 20} = \frac{50}{60} \approx 0.83$.

Step3: Find y-intercept (b)

Using the point-slope form $y = mx + b$ and substituting one point, say (20, 25): $25 = 0.83(20) + b$. Solving for b: $25 = 16.6 + b \Rightarrow b = 25 - 16.6 = 8.4$. So the equation is $y = 0.83x + 8.4$ (approximate, coefficients rounded to nearest hundredth).

Step4: Predict for x = 70

Substitute x = 70 into the equation: $y = 0.83(70) + 8.4 = 58.1 + 8.4 = 66.5$. Rounding to nearest hundredth (or nearest whole number as per context, but following the problem's instruction to round to nearest hundredth, it's 66.50, but if to nearest whole number, 67. However, let's check the line again. Maybe a better estimation: Let's take two other points, say (10, 20) and (90, 80). Then slope $m = \frac{80 - 20}{90 - 10} = \frac{60}{80} = 0.75$. Then using (10, 20): $20 = 0.75(10) + b \Rightarrow 20 = 7.5 + b \Rightarrow b = 12.5$. So equation $y = 0.75x + 12.5$. Now for x = 70: $y = 0.75(70) + 12.5 = 52.5 + 12.5 = 65$. Another estimation: Let's use the middle of the data. The line of best fit should have a positive slope. Let's take (30, 35) and (70, 70). Slope $m = \frac{70 - 35}{70 - 30} = \frac{35}{40} = 0.875$. Then $35 = 0.875(30) + b \Rightarrow 35 = 26.25 + b \Rightarrow b = 8.75$. Equation $y = 0.875x + 8.75$. For x = 70: $y = 0.875(70) + 8.75 = 61.25 + 8.75 = 70$. Since the scatter plot's line is increasing, the prediction should be around 65 - 70. Let's use the first estimated equation $y = 0.83x + 8.4$. For x = 70, $y = 0.83*70 + 8.4 = 58.1 + 8.4 = 66.5$. So approximately 66.5 (or 67 when rounded to nearest whole number).

Answer:

(for part b, assuming the equation from part a is estimated properly):
For part (a), an approximate equation could be $y = 0.75x + 12.5$ (or other reasonable estimation). For part (b), the predicted quiz score when x = 70 minutes is approximately 65 - 70, with a more precise calculation (using a better line fit) giving around 66.5 (or 67). However, following the step-by-step, if we use the first estimation with slope 0.83 and intercept 8.4, the answer for part (b) is 66.5 (or 67 when rounded to nearest whole number, but the problem says "round your prediction to the nearest hundredth", so 66.50 or 66.5).