Question

question #1
use the residual plot below to state what conclusion can be made about the regression equation.
residual plot
the regression equation is a good model because the graph appears to be more spread out on one side.
the regression equation is a good model because there appears to be no obvious pattern.
the regression equation is a bad model because the graph appears to be more spread out on one side.
the regression equation is a bad model because there appears to be no obvious pattern.
question #2
use the given data set to determine the correlation coefficient r.

Explanation:

Question #1

A good regression model has a residual plot with no obvious pattern. If there is no pattern (like no curve, no funnel - shape), the model fits the data well. A pattern (like a curve) would indicate a poor - fitting model. Also, being "spread out on one side" is not a correct criterion for a good or bad model.

Answer:

The regression equation is a good model because there appears to be no obvious pattern.

Question #2

Let \(x\) be the number of pets and \(y\) be the cost of pet food.
We have \(n = 5\) (since there are 5 data points).
First, calculate the means:
\(\bar{x}=\frac{1 + 2+3+4+5}{5}=\frac{15}{5}=3\)
\(\bar{y}=\frac{7 + 12+5+6+24}{5}=\frac{54}{5}=10.8\)

Next, calculate the following sums:
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 3)(7 - 10.8)+(2 - 3)(12 - 10.8)+(3 - 3)(5 - 10.8)+(4 - 3)(6 - 10.8)+(5 - 3)(24 - 10.8)\)
\(=(- 2)(-3.8)+(-1)(1.2)+(0)(-5.8)+(1)(-4.8)+(2)(13.2)\)
\(=7.6-1.2 + 0-4.8+26.4=28\)

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(1 - 3)^{2}+(2 - 3)^{2}+(3 - 3)^{2}+(4 - 3)^{2}+(5 - 3)^{2}\)
\(=(-2)^{2}+(-1)^{2}+0^{2}+1^{2}+2^{2}=4 + 1+0+1+4 = 10\)

\(\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=(7 - 10.8)^{2}+(12 - 10.8)^{2}+(5 - 10.8)^{2}+(6 - 10.8)^{2}+(24 - 10.8)^{2}\)
\(=(-3.8)^{2}+(1.2)^{2}+(-5.8)^{2}+(-4.8)^{2}+(13.2)^{2}\)
\(=14.44+1.44+33.64+23.04+174.24=246.8\)

The correlation coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)

Substitute the values:
\(r=\frac{28}{\sqrt{10\times246.8}}=\frac{28}{\sqrt{2468}}\approx\frac{28}{49.68}\approx0.564\)