Thank you for visiting A construction manager is monitoring the progress of building a new house The scatterplot and table below show the number of months since the start. This page is designed to guide you through key points and clear explanations related to the topic at hand. We aim to make your learning experience smooth, insightful, and informative. Dive in and discover the answers you're looking for!
Answer :
We begin with the given data points for the number of months since the start of the build ([tex]$x$[/tex]) and the percentage of the house left to build ([tex]$y$[/tex]):
[tex]$$
\begin{array}{cc}
x & y \\
\hline
0 & 100 \\
1 & 86 \\
2 & 65 \\
3 & 59 \\
4 & 41 \\
5 & 34 \\
\end{array}
$$[/tex]
Since a linear function of the form
[tex]$$
y = mx + b
$$[/tex]
is to be used to model this relationship, we need to determine the slope ([tex]$m$[/tex]) and the [tex]$y$[/tex]-intercept ([tex]$b$[/tex]).
### Step 1. Calculate the Slope
The slope is given by the formula
[tex]$$
m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2},
$$[/tex]
where [tex]$n$[/tex] is the number of data points.
1. There are [tex]$n = 6$[/tex] data points.
2. The sums needed are:
- [tex]$\sum x = 0 + 1 + 2 + 3 + 4 + 5$[/tex]
- [tex]$\sum y = 100 + 86 + 65 + 59 + 41 + 34$[/tex]
- [tex]$\sum (xy)$[/tex] is the sum of the product of each pair [tex]$(x,y)$[/tex].
- [tex]$\sum (x^2)$[/tex] is the sum of the squares of [tex]$x$[/tex].
By substituting and calculating, we determine that the slope is approximately
[tex]$$
m \approx -13.457.
$$[/tex]
### Step 2. Calculate the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept is given by
[tex]$$
b = \frac{\sum y - m\sum x}{n}.
$$[/tex]
Using the previously calculated slope and appropriate sums, we find that
[tex]$$
b \approx 97.810.
$$[/tex]
### Step 3. Compare with the Given Options
The candidate functions given are:
1. [tex]$\displaystyle y = -13.5x + 97.8$[/tex]
2. [tex]$\displaystyle y = -13.5x + 7.3$[/tex]
3. [tex]$\displaystyle y = 97.8x - 13.5$[/tex]
4. [tex]$\displaystyle y = 7.3x - 97.8$[/tex]
Our calculated slope of approximately [tex]$-13.457$[/tex] and [tex]$y$[/tex]-intercept of approximately [tex]$97.810$[/tex] closely match the values in option 1, which is
[tex]$$
y = -13.5x + 97.8.
$$[/tex]
### Conclusion
Since the computed values are nearly identical to those in the first candidate function, the linear function that best models the data is:
[tex]$$
\boxed{y=-13.5x+97.8.}
$$[/tex]
[tex]$$
\begin{array}{cc}
x & y \\
\hline
0 & 100 \\
1 & 86 \\
2 & 65 \\
3 & 59 \\
4 & 41 \\
5 & 34 \\
\end{array}
$$[/tex]
Since a linear function of the form
[tex]$$
y = mx + b
$$[/tex]
is to be used to model this relationship, we need to determine the slope ([tex]$m$[/tex]) and the [tex]$y$[/tex]-intercept ([tex]$b$[/tex]).
### Step 1. Calculate the Slope
The slope is given by the formula
[tex]$$
m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2},
$$[/tex]
where [tex]$n$[/tex] is the number of data points.
1. There are [tex]$n = 6$[/tex] data points.
2. The sums needed are:
- [tex]$\sum x = 0 + 1 + 2 + 3 + 4 + 5$[/tex]
- [tex]$\sum y = 100 + 86 + 65 + 59 + 41 + 34$[/tex]
- [tex]$\sum (xy)$[/tex] is the sum of the product of each pair [tex]$(x,y)$[/tex].
- [tex]$\sum (x^2)$[/tex] is the sum of the squares of [tex]$x$[/tex].
By substituting and calculating, we determine that the slope is approximately
[tex]$$
m \approx -13.457.
$$[/tex]
### Step 2. Calculate the [tex]$y$[/tex]-intercept
The [tex]$y$[/tex]-intercept is given by
[tex]$$
b = \frac{\sum y - m\sum x}{n}.
$$[/tex]
Using the previously calculated slope and appropriate sums, we find that
[tex]$$
b \approx 97.810.
$$[/tex]
### Step 3. Compare with the Given Options
The candidate functions given are:
1. [tex]$\displaystyle y = -13.5x + 97.8$[/tex]
2. [tex]$\displaystyle y = -13.5x + 7.3$[/tex]
3. [tex]$\displaystyle y = 97.8x - 13.5$[/tex]
4. [tex]$\displaystyle y = 7.3x - 97.8$[/tex]
Our calculated slope of approximately [tex]$-13.457$[/tex] and [tex]$y$[/tex]-intercept of approximately [tex]$97.810$[/tex] closely match the values in option 1, which is
[tex]$$
y = -13.5x + 97.8.
$$[/tex]
### Conclusion
Since the computed values are nearly identical to those in the first candidate function, the linear function that best models the data is:
[tex]$$
\boxed{y=-13.5x+97.8.}
$$[/tex]
Thank you for reading the article A construction manager is monitoring the progress of building a new house The scatterplot and table below show the number of months since the start. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!
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Rewritten by : Jeany
