To determine the best regression model for the provided data, let's compare the given models with the actual number of days each year that air quality in San Diego did not meet federal standards.
Here is the provided data:
[tex]\[
\begin{array}{|c|c|}
\hline
\text{Years since 1989} & \text{Number of Days} \\
\hline
1 & 39 \\
\hline
2 & 27 \\
\hline
3 & 19 \\
\hline
4 & 14 \\
\hline
5 & 9 \\
\hline
6 & 12 \\
\hline
7 & 2 \\
\hline
8 & 1 \\
\hline
\end{array}
\][/tex]
Now let us look at the values each model predicts for the number of days.
### Power Model
The power model equation is:
[tex]\[ y = 41.21x^{-0.86} \][/tex]
The values predicted by this model are:
[tex]\[ \{41.21, 22.705, 16.021, 12.509, 10.325, 8.827, 7.731, 6.892\} \][/tex]
### Exponential Model
The exponential model equation is:
[tex]\[ y = 54.97(0.71)^x \][/tex]
The values predicted by this model are:
[tex]\[ \{39.029, 27.710, 19.674, 13.969, 9.918, 7.042, 5.000, 3.550\} \][/tex]
### Logarithmic Model
The logarithmic model equation is:
[tex]\[ y = 39.14 - 17.93 \ln x \][/tex]
The values predicted by this model are:
[tex]\[ \{39.14, 26.712, 19.442, 14.284, 10.283, 7.014, 4.250, 1.856\} \][/tex]
### Quadratic Model
The quadratic model equation is:
[tex]\[ y = 0.60x^2 - 10.38x + 46.73 \][/tex]
The values predicted by this model are:
[tex]\[ \{36.95, 28.37, 20.99, 14.81, 9.83, 6.05, 3.47, 2.09\} \][/tex]
Next, let's summarize these values:
[tex]\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Years since 1989} & \text{Actual Days} & \text{Power Model} & \text{Exponential Model} & \text{Logarithmic Model} & \text{Quadratic Model} \\
\hline
1 & 39 & 41.21 & 39.029 & 39.14 & 36.95 \\
\hline
2 & 27 & 22.705 & 27.710 & 26.712 & 28.37 \\
\hline
3 & 19 & 16.021 & 19.674 & 19.442 & 20.99 \\
\hline
4 & 14 & 12.509 & 13.969 & 14.284 & 14.81 \\
\hline
5 & 9 & 10.325 & 9.918 & 10.283 & 9.83 \\
\hline
6 & 12 & 8.827 & 7.042 & 7.014 & 6.05 \\
\hline
7 & 2 & 7.731 & 5.000 & 4.250 & 3.47 \\
\hline
8 & 1 & 6.892 & 3.550 & 1.856 & 2.09 \\
\hline
\end{array}
\][/tex]
By comparing the predicted values with the actual data, we find that:
- The Power Model tends to overestimate the number of days, especially in the middle years.
- The Exponential Model also overestimates initially but aligns quite well for the first few years.
- The Logarithmic Model closely matches the actual values throughout the years.
- The Quadratic Model matches closely for years 2 to 5 but diverges more significantly in other years.
Given this comparison, the Logarithmic Model best matches the actual data values over the range of years provided. Therefore, the phrase that best describes the regression model for the data is:
Logarithmic model; [tex]\(y = 39.14 - 17.93 \ln x\)[/tex]
