Question

scratch area (not part of answer)
a scatter plot shows the time spent studying, x, and the quiz score, y, for each of 25 students. use the scatter plot to answer the parts below. (note that you can use the graphing tools to help you approximate the line.)
(a) write an approximate equation of the line of best fit. round the coefficients to the nearest hundredth.
( y = ) ( x )
(b) using your equation from part (a), predict the quiz score for a student who spent 50 minutes studying. round your prediction to the nearest hundredth.
90.00

Explanation:

Part (a)

Step1: Identify two points on the line

Looking at the scatter plot, we can estimate two points on the line of best fit. Let's take (20, 20) and (80, 80) as approximate points (these are estimates from the trend).

Step2: Calculate the slope (m)

The slope formula is \( m = \frac{y_2 - y_1}{x_2 - x_1} \). Using the points (20, 20) and (80, 80):
\( m = \frac{80 - 20}{80 - 20} = \frac{60}{60} = 1 \)

Step3: Determine the y-intercept (b)

Using the point (20, 20) and the slope \( m = 1 \) in the equation \( y = mx + b \):
\( 20 = 1 \times 20 + b \)
\( 20 = 20 + b \)
\( b = 0 \)
So the equation of the line of best fit is \( y = 1.00x + 0.00 \) (rounded to the nearest hundredth).

Part (b)

Step1: Use the equation from part (a)

The equation is \( y = x \) (since \( m = 1 \) and \( b = 0 \)).

Step2: Substitute \( x = 50 \) into the equation

\( y = 50 \) (since \( y = x \), when \( x = 50 \), \( y = 50 \)). But wait, maybe our initial points were off. Let's re - estimate the line of best fit more accurately. Let's take two other points, say (10, 15) and (90, 85).

Step1 (re - estimating slope):

Slope \( m=\frac{85 - 15}{90 - 10}=\frac{70}{80}=0.875\approx0.88 \)

Step2 (re - estimating y - intercept):

Using point (10, 15) and \( m = 0.88 \) in \( y=mx + b \)
\( 15=0.88\times10 + b \)
\( 15 = 8.8 + b \)
\( b=15 - 8.8 = 6.2\approx6.20 \)
So the equation is \( y = 0.88x+6.20 \)

Step3: Substitute \( x = 50 \) into the new equation

\( y=0.88\times50 + 6.20 \)
\( y = 44+6.20=50.20 \)

Answer:

(Part a):
\( y = 0.88x + 6.20 \) (this is a more accurate estimate, the first estimate was a special case. The actual trend from the scatter plot shows a positive linear relationship with slope less than 1 when estimated more carefully)