Question

a linear model is y = -0.3x + 91.9

1. which statement best represents the interpretation of the slope?

i. the math test score decreases by approximately 6 points for every additional hour spent watching tv.
ii. the math test score increases by approximately 6 points for every additional hour spent watching tv.
iii. the math test score decreases by 1 point for approximately every 6 hours of watching tv.

Explanation:

Step1: Recall Slope Interpretation

The linear model is \( y = -0.3x + 91.9 \) (assuming the slope is -0.3, maybe a typo in the original, but for slope interpretation: in \( y = mx + b \), \( m \) is slope, representing change in \( y \) per unit \( x \). If we assume the slope is -0.3 (close to -1/3, but maybe the options have a typo, or original slope was -1/6? Wait, the options talk about 6 or 1. Wait, maybe the original equation was \( y = - \frac{1}{6}x + 91.9 \) (slope -1/6 ≈ -0.166, no). Wait, the options: let's re-express. If slope is -0.3, no. Wait, maybe the equation is \( y = - \frac{1}{6}x + 91.9 \), but the options:

Option I: Decrease by ~1 per hour TV? No. Wait, maybe the slope is -1/6, so for 6 hours TV, y decreases by 1. Wait, option III: "decreases by 1 point for approximately every 6 hours of watching TV" – that would mean slope is -1/6, since change in y (1 decrease) over change in x (6 hours) is -1/6.

Wait, let's re-express the slope interpretation: in \( y = mx + b \), \( m \) is \( \frac{\Delta y}{\Delta x} \). So if \( m = - \frac{1}{6} \), then for \( \Delta x = 6 \), \( \Delta y = -1 \) (decrease by 1). So option III: "The math test score decreases by 1 point for approximately every 6 hours of watching TV" matches, because when x (hours TV) increases by 6, y (score) decreases by 1.

Option I: If slope was -1, then y decreases by 1 per x=1. Option II: positive slope, but our slope is negative (since equation has -0.3 or -1/6, negative). So II is wrong (increase). I: if slope -1, but our slope is not -1. III: slope -1/6, so for x=6, y decreases by 1. So III is correct.

Step2: Eliminate Wrong Options

• Option II: slope is negative (decrease), so "increases" is wrong. Eliminate II.
• Option I: If slope was -1, but our slope (from options' context) is -1/6 (since III is about 6 hours for 1 decrease). So I's "decreases by ~1 per hour" would need slope -1, which doesn't match. Eliminate I.
• Option III: Matches slope interpretation: \( \Delta y = -1 \) when \( \Delta x = 6 \), so slope \( m = \frac{-1}{6} \), which fits the idea of decreasing by 1 over 6 hours.

Answer:

III (The math test score decreases by 1 point for approximately every 6 hours of watching TV)