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Following are age and price data for 8 randomly selected ambulances between 1 and 6 years old. Here, [tex]$x$[/tex] denotes age, in years, and [tex]$y$[/tex] denotes price, in hundreds of dollars. Use the information to do parts (a) through (d).

[tex]$
\begin{array}{l}
\begin{array}{ccccccccc}
x & 6 & 1 & 6 & 2 & 6 & 2 & 4 & 5 \\
\hline y & 260 & 400 & 255 & 350 & 265 & 360 & 310 & 295
\end{array} \\
\sum x=32, \sum y=2495, \sum x y=9215, \sum x^2=158, \sum y^2=798075 \\
\end{array}
$[/tex]

a. Compute SST, SSR, and SSE using the formulas:

[tex]\[
\text{SST} = \sum y_i^2 - \left( \frac{(\sum y_i)^2}{n} \right)
\][/tex]

[tex]\[
\text{SSR} = \frac{\left( \sum x_i y_i - \frac{(\sum x_i)(\sum y_i)}{n} \right)^2}{\sum x_i^2 - \frac{(\sum x_i)^2}{n}}
\][/tex]

[tex]\[
\text{SSE} = \text{SST} - \text{SSR}
\][/tex]

[tex]\[
\text{SST} = \square \text{ (Round to two decimal places as needed.) }
\][/tex]

Sagot :

GinnyAnsweridn
Certainly! Let's tackle each part step-by-step using the given formulas and data.

### Given Data:
[tex]\[ \begin{array}{cc} x & y \\ \hline 6 & 260 \\ 1 & 400 \\ 6 & 255 \\ 2 & 350 \\ 6 & 265 \\ 2 & 360 \\ 4 & 310 \\ 5 & 295 \\ \end{array} \][/tex]
Additional provided sums:
[tex]\[ \sum x = 32, \sum y = 2495, \sum xy = 9215, \sum x^2 = 158, \sum y^2 = 798075 \][/tex]

The sample size [tex]\(n\)[/tex] is 8.

### Step-by-Step Solution

#### Part (a): Compute SST, SSR, and SSE

1. Calculate SST (Total Sum of Squares):

The formula for SST is:
[tex]\[ \text{SST} = \sum y_i^2 - \frac{(\sum y_i)^2}{n} \][/tex]

Plugging in the provided values:
[tex]\[ \text{SST} = 798075 - \frac{(2495)^2}{8} \][/tex]

First, compute the squared term:
[tex]\[ 2495^2 = 6225025 \][/tex]

Then divide by [tex]\(n\)[/tex]:
[tex]\[ \frac{6225025}{8} = 778128.125 \][/tex]

Finally, calculate SST:
[tex]\[ \text{SST} = 798075 - 778128.125 = 19946.88 \][/tex]

So, SST = 19946.88.

2. Calculate SSR (Regression Sum of Squares):

The formula for SSR is:
[tex]\[ \text{SSR} = \frac{\left( \sum xy - \frac{(\sum x)(\sum y)}{n} \right)^2}{\sum x^2 - \frac{(\sum x)^2}{n}} \][/tex]

First, compute [tex]\(\sum xy - \frac{(\sum x)(\sum y)}{n}\)[/tex]:
[tex]\[ \sum xy - \frac{(\sum x)(\sum y)}{n} = 9215 - \frac{32 \times 2495}{8} \][/tex]

Compute the product term:
[tex]\[ 32 \times 2495 = 79840 \][/tex]

Then divide:
[tex]\[ \frac{79840}{8} = 9980 \][/tex]

Then:
[tex]\[ 9215 - 9980 = -765 \][/tex]

Next, compute [tex]\(\sum x^2 - \frac{(\sum x)^2}{n}\)[/tex]:
[tex]\[ \sum x^2 - \frac{(\sum x)^2}{n} = 158 - \frac{32^2}{8} \][/tex]

Compute the squared term:
[tex]\[ 32^2 = 1024 \][/tex]

Then divide:
[tex]\[ \frac{1024}{8} = 128 \][/tex]

Then:
[tex]\[ 158 - 128 = 30 \][/tex]

Finally, calculate SSR:
[tex]\[ \text{SSR} = \frac{(-765)^2}{30} = \frac{585225}{30} = 19507.5 \][/tex]

So, SSR = 19507.5.

3. Calculate SSE (Error Sum of Squares):

The formula for SSE is:
[tex]\[ \text{SSE} = \text{SST} - \text{SSR} \][/tex]

Substituting the values:
[tex]\[ \text{SSE} = 19946.88 - 19507.5 = 439.38 \][/tex]

So, SSE = 439.38.

### Summary

[tex]\[ \begin{aligned} \text{SST} & = 19946.88 \\ \text{SSR} & = 19507.5 \\ \text{SSE} & = 439.38 \\ \end{aligned} \][/tex]

These values are rounded to two decimal places as requested.
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