Question

The mean number of students per classroom, y, at Central High School can be estimated using the equation\[y = 0.8636x + 27.227\], where $x$ represents the number of years since 2004 and \[x \leqslant 10\]. Which of the following statements is the best interpretation of the number 0.8636 in the context of this problem?
A) The estimated mean number of students per classroom in 2004
B) The estimated mean number of students per classroom in 2014
C) The estimated yearly decrease in the mean number of students per classroom
D) The estimated yearly increase in the mean number of students per classroom

Answer 1

Hint:Here interpret the slope of the equation compared in relation to the real-world scenario.Additionally, when the models are created from data, one must identify that these models only and therefore eliminate the independent variable $y$, for the given value of $x$.

Complete step-by-step solution
Compare the given equation \[y = 0.8636x + 27.227\] with the standard form \[y = mx + b\;\] we get,
$
  m = 0.8636 \\
  b = 27.227 \\
 $
>Option A is not the correct answer as it is clearly mentioned in the question that it is the year in which the mean of the students is measured and as the year is changing, so we cannot say 0.8636 is the estimated mean number of students per classroom in 2004.
>Option B is not the correct answer as it is clearly mentioned in the question that it is the year in which the mean of the students is measured. Moreover, 2014 is not even in the question; instead, 2004 is mentioned.
>Option C is not the correct answer as the slope of the equation is positive (>0) and so, estimated yearly increases in the mean number of students per classroom.
>The slope of a linear equation gives the amount that the mean is the number of students per classroom that is represented by $y$ changes per year which are represented by $x$. So, 0.8636 is the estimated yearly increase in the mean number of students per classroom.

Hence, option D is the correct answer.

Note:The mean is the average of the numbers. It is easy to calculate: the sum of all the numbers, then divide by how many numbers there are. In other words, it is the sum divided by the count.