Question

cool down: roar of the crowd
the scatter plot shows the maximum noise level when different numbers of people are in a stadium. the linear model is given by the equation ( y = 1.5x + 22.7 ), where ( y ) represents maximum noise level and ( x ) represents the number of people, in thousands, in the stadium.

1. the slope of the linear model is 1.5. what does this mean in terms of the maximum noise level and the number of people?
2. sports announcer states that there are 65,000 fans in the stadium. estimate the maximum noise level. is this estimate reasonable? explain your reasoning.
3. what is the ( y )-intercept of the linear model given? what does it mean in the context of the problem? is this reasonable? explain your reasoning.

Explanation:

1.

The slope in a linear equation \(y = mx + b\) (here \(y = 1.5x+22.7\)) represents the rate of change. For each additional 1000 people (\(x\) increases by 1), the maximum noise level (\(y\)) increases by \(1.5\) decibels.

For the linear equation \(y=1.5x + 22.7\), the \(y\) - intercept is \(b = 22.7\). In the context of the problem, when \(x = 0\) (i.e., there are \(0\) people in the stadium), \(y=22.7\). But in reality, if there are \(0\) people in the stadium, there should be no crowd - related noise. However, it could represent some base noise level (e.g., from stadium equipment)

Answer:

For every 1000 - person increase in the stadium crowd, the maximum noise level increases by \(1.5\) decibels.

2.

Step - by - Step Format:

Step1: Substitute \(x = 65\) into the equation

Given \(y=1.5x + 22.7\), when \(x = 65\) (since \(x\) is in thousands), we substitute \(x\) into the equation:
\(y=1.5\times65+22.7\)

Step2: Calculate the value of \(y\)

First, calculate \(1.5\times65=97.5\). Then \(y=97.5 + 22.7=120.2\)

Step3: Check reasonableness

Looking at the scatter - plot, when \(x = 65\), the data points are around the value predicted by the line \(y = 1.5x+22.7\). The line is a good fit for the data points in the vicinity of \(x = 65\).