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:

Step1: Interpret the slope

The slope represents the rate of change. For every 1 - thousand - person increase in the number of people in the stadium ($x$ increases by 1), the maximum noise level ($y$) increases by 1.5 decibels.

Step2: Estimate the noise level for 65,000 people

Since $x$ is in thousands, for 65,000 people, $x = 65$. Substitute $x = 65$ into the equation $y=1.5x + 22.7$.
$y=1.5\times65+22.7$
$y = 97.5+22.7$
$y=120.2$ decibels. This estimate is reasonable as the point $(65, 120.2)$ lies close to the trend of the scatter - plot data points.

Step3: Find and interpret the y - intercept

The equation is in the form $y=mx + b$, where $b$ is the y - intercept. Here, $b = 22.7$. In the context of the problem, it means that when there are 0 people in the stadium ($x = 0$), the maximum noise level is 22.7 decibels. This may not be entirely reasonable in a strict sense as a stadium with no people should have a noise level close to the ambient noise level of the empty stadium, but it could represent some base - level noise from stadium equipment or other non - human sources.

Answer:

1. For every 1 - thousand - person increase in the number of people in the stadium, the maximum noise level increases by 1.5 decibels.
2. The estimated maximum noise level is 120.2 decibels. This estimate is reasonable as it lies close to the trend of the data points in the scatter - plot.
3. The y - intercept is 22.7. It means that when there are 0 people in the stadium, the maximum noise level is 22.7 decibels. It may not be entirely reasonable as a stadium with no people should have a very low noise level, but could represent some base - level non - human noise.