To determine the number of employees in the cellular telephone industry in the United States in 1994 (4 years since 1990), we can use polynomial regression to fit a second-degree polynomial (quadratic) to the given data points. Let's walk through the step-by-step process for finding this solution.
1. Data Points:
We have the following data points for the years since 1990 and the corresponding number of employees:
[tex]\[
\begin{array}{|c|c|}
\hline \text{Years Since 1990} & \text{Employees} \\
\hline 0 & 21,400 \\
\hline 1 & 26,300 \\
\hline 2 & 34,300 \\
\hline 3 & 39,800 \\
\hline 5 & 68,200 \\
\hline
\end{array}
\][/tex]
2. Form of Polynomial:
We will fit a quadratic polynomial of the form:
[tex]\[
E(x) = ax^2 + bx + c
\][/tex]
where [tex]\( E(x) \)[/tex] is the number of employees, and [tex]\( x \)[/tex] is the number of years since 1990.
3. Finding Coefficients:
By fitting the given data points to the quadratic polynomial, we determine the coefficients [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. These coefficients are found to be:
[tex]\[
a = 1282.4006, \quad b = 2761.0457, \quad c = 21922.975
\][/tex]
4. Constructing the Polynomial:
Using the determined coefficients, the polynomial representing the data is:
[tex]\[
E(x) = 1282.4006x^2 + 2761.0457x + 21922.975
\][/tex]
5. Estimating Employees in 1994:
To find the number of employees in 1994, we substitute [tex]\( x = 4 \)[/tex] into the polynomial:
[tex]\[
E(4) = 1282.4006 \cdot 4^2 + 2761.0457 \cdot 4 + 21922.975
\][/tex]
Calculating this step-by-step:
[tex]\[
4^2 = 16
\][/tex]
[tex]\[
1282.4006 \cdot 16 = 20518.4096
\][/tex]
[tex]\[
2761.0457 \cdot 4 = 11044.1828
\][/tex]
[tex]\[
E(4) = 20518.4096 + 11044.1828 + 21922.975
\][/tex]
[tex]\[
E(4) = 53485.567
\][/tex]
6. Result:
Therefore, the estimated number of employees in the cellular telephone industry in 1994 is approximately [tex]\( 53,485.57 \)[/tex].
Thus, the number of employees in the cellular telephone industry in the United States in 1994 is approximately 53,485.
