ihav2angels
Answered

The table shows data from local day-care centers, representing the number of children in attendance [tex]$(x)$[/tex] and daily food costs in dollars ([tex]$y$[/tex]).

\begin{tabular}{|c|c|c|c|}
\hline
[tex]$x$[/tex] & [tex]$y$[/tex] & [tex]$x^2$[/tex] & [tex]$x y$[/tex] \\
\hline
16 & 45 & 256 & 720 \\
\hline
22 & 58 & 484 & 1,276 \\
\hline
28 & 73 & 784 & 2,044 \\
\hline
32 & 94 & 1,024 & 3,008 \\
\hline
45 & 141 & 2,025 & 6,345 \\
\hline
[tex]$\sum x=143$[/tex] & [tex]$\sum y=411$[/tex] & [tex]$\sum x^2=4,573$[/tex] & [tex]$\sum x y=13,393$[/tex] \\
\hline
\end{tabular}

Which regression equation correctly models the data?

A. [tex]$y = 2.87x + 0.12$[/tex]
B. [tex]$y = 2.87x + 11.85$[/tex]



Answer :

GinnyAnswer
To determine the correct regression equation that models the data, we need to find the equation of the form [tex]\( y = ax + b \)[/tex], where [tex]\( a \)[/tex] is the intercept and [tex]\( b \)[/tex] is the slope.

Given the following sums from the data table:

[tex]\[ \begin{aligned} & \sum x = 143 \\ & \sum y = 411 \\ & \sum x^2 = 4573 \\ & \sum xy = 13393 \\ & n = 5 \quad \text{(number of data points)} \end{aligned} \][/tex]

We can calculate the slope ([tex]\( b \)[/tex]) and intercept ([tex]\( a \)[/tex]) of the regression line using the formulas:

[tex]\[ b = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2} \][/tex]

[tex]\[ a = \frac{(\sum y)(\sum x^2) - (\sum x)(\sum xy)}{n(\sum x^2) - (\sum x)^2} \][/tex]

Substituting the given values:

1. Calculate the slope ([tex]\( b \)[/tex]):
[tex]\[ b = \frac{5 \cdot 13393 - 143 \cdot 411}{5 \cdot 4573 - 143^2} \][/tex]

2. Calculate the intercept ([tex]\( a \)[/tex]):
[tex]\[ a = \frac{411 \cdot 4573 - 143 \cdot 13393}{5 \cdot 4573 - 143^2} \][/tex]

After evaluating these expressions, we find the following approximate values:

[tex]\[ a \approx -14.77 \][/tex]

[tex]\[ b \approx 3.39 \][/tex]

Thus, the regression equation that correctly models the given data is:

[tex]\[ y = 3.39x - 14.77 \][/tex]

Comparing this with the given options:

- [tex]\( y = 2.87x + 0.12 \)[/tex]
- [tex]\( y = 2.87x + 11.85 \)[/tex]

Neither of these options match the calculated regression equation, indicating that provided options of [tex]\( y = 2.87x + 0.12 \)[/tex] and [tex]\( y = 2.87x + 11.85 \)[/tex] are incorrect. The correct regression equation based on the given data is:

[tex]\[ y = 3.39x - 14.77 \][/tex]

Other Questions